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ELEMENTARY    ALGEBRA 


J.  A.  GILLET 

Professor  in  the  New  York  Normal  College 


NEW  YORK 
HENRY  HOLT  AND  COMPANY 

1896 


•^  *  .*.         •    •  *  i'  «•> «  s 


Copyright,  1896, 

BY 

HENRY  HOLT  &  CO. 


ROBERT  DRUMMOND,    EI.ECTROTYPER    ANP    PRINTER,   NEW   YORK. 


.^c 


PREFACE. 


This  book  is  designed  to  be  at  once  simple  enough  for 
the  beginner  and  complete  enough  for  the  most  advanced 
classes  of  academies  and  preparatory  schools.  The  first 
three  quarters  of  it  constitute  an  elementary  algebra  in  the 
strictest  sense  of  the  term;  the  remainder  may  be  regarded 
as  an  intermediate  step  between  elementary  and  higher 
algebra,  and  includes  the  topics  of  the  most  advanced  re- 
quirements in  this  subject  for  admission  to  American  col- 
leges and  technical  schools. 

One  of  the  main  differences  between  this  book  and  its 
American  predecessors  lies  in  the  prominence  given  to 
problems  and  the  consequent  early  introduction  of  the  equa- 
tion. The  statement  of  problems  in  the  form  of  equations 
calls  forth  the  pupil's  intellectual  resources  and  develops  in 
him  the  power  of  concentrated  thought.  It  is  an  invalua- 
ble mental  exercise,  and  one,  moreover,  in  which  as  a  rule 
pupils  take  pleasure.  Drill  in  algebraic  operations,  on  the 
other  hand,  tends  rather  to  strengthen  the  memory,  to 
quicken  the  apprehension,  and  to  cultivate  habits  of  ac- 
curacy. Though  absolutely  necessary  to  secure  facility  in 
manipulating  algebraic  expressions,  this  drill  is  apt  not  to 
be  interesting.  For  the  sake,  therefore,  both  of  giving 
varied  employment  to  the  mental  activities  and  of  main- 
taining an  equilibrium  of  interest,  it  seems  desirable  that 

ill 

800537 


iv  PREFACE. 

problems  and  exercises  should  proceed  together  from  the 
very  outset.  Problems  are  accordingly  introduced  at  a 
much  earlier  stage  than  usual,  and  occur  with  uncommon 
frequency  in  every  chapter.  At  first  they  are  so  simple 
that  the  resulting  equations  can  be  solved  by  elementary 
arithmetical  processes,  and  they  gradually  increase  in  com- 
plication with  the  pupil's  increasing  knowledge  of  algebraic 
methods.  The  majority  of  them  are  either  new  or  else  the 
old  ones  with  new  data;  the  remainder  have  been  selected 
from  a  great  variety  of  sources. 

The  book  further  differs  from  its  predecessors  (1)  in 
the  attention  given  to  negative  quantities  and  to  the  formal 
laws  of  algebra,  known  as  the  Commutative,  the  Associa- 
tive, the  Distributive,  and  the  Index  laws.  In  presenting 
these  laws  the  author  has  endeavored  to  be  rigorous  without 
sacrificing  simplicity.  (2)  In  the  fuller  development  of 
factoring  and  in  its  more  extensive  application  to  the  solu- 
tion of  equations.  The  method  of  solving  quadratic  equa- 
tions has  been  based  entirely  on  the  principles  of  factoring. 
Certainly  this  method  is  more  in  harmony  with  the  pro- 
cesses of  advanced  algebra,  and  it  is  the  author's  experience 
that,  even  for  the  beginner,  it  is  quite  as  simple  as  the 
method  of  completing  the  square. 

The  first  steps  in  the  book  have  been  simplified  for  the 
pupil  by  building  upon  his  knowledge  of  arithmetic  and 
adding,  one  by  one,  the  distinguishing  features  of  algebra; 
— the  use  of  letters  as  well  as  figures  to  express  numbers, 
the  use  of  equations  in  the  solution  of  problems,  the  more 
extended  and  systematic  use  of  signs,  the  meaning  and  use 
of  negative  numbers,  and  the  general  proof  of  theorems. 
In  further  recognition  of  practical  requirements,  the  exer- 
cises in  Part  I  have  been  divided  usually  into  two  sets,  the 
first  set  being  as  a  rule  easier  than  the  second.  Careful 
provision  is  made  in  both  sets  for  frequent  review  of  topics 
already  studied. 


PREFACE.  V 

As  the  author  and  publisher  cannot  hope  to  have  been 
entirely  successful  in  their  efforts  to  keep  the  text  free 
from  typographical  and  other  errors,  they  will  esteem  it  a 
favor  to  have  their  attention  called  to  any  that  may  have 
escaped  their  vigilance. 

J.  A.  G. 
Normal  College,  New  York, 
December  10,  1895. 


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TABLE   OP  CONTENTS. 

(The  numbers  refer  to  the  pages  of  the  text.) 

PART   I. 

FUNDAMENTAL    PRINCIPLES    AND    OPERATIONS- 
CHAPTER  I. 
NOTATION  AND  SYMBOLS. 

Symbols  of  Operation — Algebraic  Expressions,  1. — Exponents — Co- 
efficients, 2. — Numeric  Values,  3. — Quantitative  Symbols  — 
Terms — Monomials  and  Polynomials,  4.— Similar  Terms,  5. 

CHAPTER   II. 

EQUATIONS  AND  PARENTHESES. 

Members  of  an  Equation — Verbal  Symbols — Axioms,  7. — Transposi- 
tion of  Terms — Collection  of  Terms— Division  by  Coefficient  of  x 
— Solution  of  an  Equation,  8. — Literal  Coefficients — Solution  of 
Problems,  9. — Clearing  of  Fractions,  11. — Symbols  of  Aggrega- 
tion, 14. — Signs  of  Parentheses,  15. — Parenthetic  Factors,  16. — 
Note  on  Transposition,  18. 

CHAPTER  III. 

NEGATIVE  QUANTITIES. 

Counting — Signs  of  Quality,  21. — Scale  of  Numbers,  22. — Absolute 
and  Actual  Values — Addition  and  Subtraction  of  Integers,  23. — 
(Corresponding  Positive  and  Negative  Numbers — Special  Signs  of 
Quality,  25. — C'ommutative  Law  of  Addition — Addition  and  Sub- 
traction of  Corresponding  Numbers,  26. — Associative  Law  of 
Addition — Oppositeness  of  Positive  and  Negative  Numbers,  27. 

CHAPTER  IV. 

ADDITION  OF  INTEGRAL  EXPRESSIONS. 

Arithmetical  and  Algebraic  Sums,  32. — Signs  of  Coefficients — Inte- 
gral Expressions — Extension  of  the  Formal  Laws  of  Addition — 
Definition  of  Addition  of  Algebraic  Expressions,  33. — Addition 
of  Monomials  and  Polynomials,  34. — Simplification  of  Polyno- 
mials, 36. — Aggregation  of  Coefficients,  38. 

vii 


Viii  TABLE  OF  CONTENTS. 

CHAPTER  V. 

SUBTRACTION  OF  INTEGRAL  EXPRESSIONS. 

Definition  of  Subtraction — Rule  for  Subtraction  of  Integral  Expres- 
sions, 41. — Operations  on  Aggregates,  43. — Compound  Paren- 
theses, 45. 

CHAPTER  VI. 

MULTIPLICATION  OF  INTEGRAL  EXPRESSIONS. 

Multiplication  of  Integers — Two  Cases  of  Multiplication,  49, — Law  of 
(Signs  in  Multiplication — Commutation  Law  of  Multiplication, 
50. — Associative  Law  of  Multiplication,  51. — Multiplication  of 
Monomials,  52. — Changing  the  Signs  of  an  Equation,  54. — Dis- 
tributive Law  of  Multiplication  of  Integers,  55. — Extension  of 
the  Distributive  Law,  59. — Arrangement  of  Terms  according  to 
the  Powers  of  a  Letter — Multiplicati(m  of  Polynomials,  61. — 
Multiplication  by  Detached  Coefficients,  64. — Degree  of  an  In- 
tegral Expression,  66. — Product  of  Homogeneous  Expressions — 
Highest  and  Lowest  Terms  of  a  Product,  67  — Complete  and  In- 
complete Integral  Expressions,  68. 

CHAPTER  VII. 

DIVISION  OF  INTEGRAL  EXPRESSIONS. 

Definition  of  Division — Division  of  Monomials,  69. — Division  of 
Polynomials,  71. — Freeing  an  Equation  from  Expressions  of  Di- 
vision, 77. — Synthetic  Division,  79. 

CHAPTER  Vni. 

INVOLUTION  OF  INTEGRAL  EXPRESSIONS. 

Definition  of  Involution— Involution  of  Monomials,  87. — Squaring  of 
Binomials,  88. — Squaring  of  Polynomials,  89. — Cubing  of  Bino- 
mials, 90. 

CHAPTER  IX. 

EVOLUTION  OF  INTEGRAL  EXPRESSIONS. 

Definition  of  Evolution — Inverse  of  Involution,  93. — Corresponding 
Direct  and  Inverse  Operations  do  not  always  cancel,  94. — Extrac- 
tion of  Root  of  Monomials,  95. — Square  Root  of  Polynomials, 
96. — Squaring  Numbers  as  Polynomials,  97. — Square  Root  of 
Numbers,  99. — Cubing  of  Polynomials,  103. — Cube  Root  of 
Polynomials,  103.— Cubing  Numbers  as  Polynomials,  104. — 
Cube  Root  of  Numbers,  105. 

CHAPTER  X. 

MULTIPLICATION  AT  SIGHT. 

Complete  Expression  of  the  First  and  Second  Degree — Product  of  Two 
Linear  Binomials,  108. — Product  of  x-\-  a  and  x  -\-  b — Product  of 


TABLE  OF  CONTENTS.  IX 

x-\-a  and  x  -{-  a — Product  ot  x-\-  a  and  x  —  a,  110.— Product  of 
«a;4-&and  cx-\-d,  111.— Product  of  Binomial  Aggregates,  112. 
— Product  of  X  4"  y  and  x^  —  xy  -f-  y'^ — Product  of  x  —  y  and 
x^  ■-{- xy -\- y'\  114. — To  con  vert  ;r- -|- ^^  ii^to  a  Perfect  Square, 
115. — To  convert  x^'^  +  ^-c"  into  a  Perfect  Square,  116. — To  con- 
vert x^  -\-bx-\-c  into  a  Perfect  Square,  117. — To  convert  ax^-\-bx 
into  a  Perfect  Square,  118. 

CHAPTER  XI. 

FACTORING. 

Resolution  of  an  Expression  into  Factors — Resolution  of  an  Expression 
in  Monomial  and  Polynomial  Factors,  120. — To  factor  the  Differ- 
ence of  Two  Squares,  121. — Special  Cases  of  factoring  Quadratic 
Trinomials,  122, — Functions,  124. — Remainder  Theorem,  126. — 
To  factor  the  Sum  and  Difference  of  the  Same  Powers  of  Two 
Quantities,  129. 

CHAPTER  XII. 

HIGHEST  COMMON  FACTORS. 

Definition  of  Highest  Common  Factor — H.C.F.  of  Monomials,  132. — 
H.  C.  F.  of  Polynomials  by  Inspection,  133. — General  Method  of 
finding  the  Highest  Common  Factor  of  Polynomials,  135. — Gen- 
eral Method  for  Three  or  More  Polynomials— H.C.F.  not  neces- 
sarily the  G.  C.  M.,  139. 

CHAPTER  XIII. 

LOWEST  COMMON  MULTIPLE. 

Definition  of  Lowest  Common  Multiple — L.  C.  M.  by  Inspection,  144. 
— L.  C.  M.  by  Division,  145. 

CHAPTER  XIV. 

FRACTIONS. 

The  Symbol  — ,  150. — The  Denominator  of  a  Fraction  is  Distributive, 

151. — Theorem  :  —  =  — r-,  loo. — Iheorem  :  --  z=  , Sim- 

h       mh  b       h  -i-  m 

plification  of  Fractions, 154. — Reduction  of  Fractions  to  a  Common 

Denominator,  156. — Theorem  :  i-  X  -7  =  rr.    158.  —  Corollary  : 

b       d       bd  '' 

-  X  c  —  c  X  -r  =  -fy  159. — Reciprocal  of  a  Fraction — Theorem  : 

a        c       ad  a  a  a^e     ... 

X Corollary  :  -    -i-  c  =    —  =  — - — ,   161. — 


b       d        b         c  b 

Corollary  :  c  -i-  —  =  c  X  - ,  162 

b  a 

of  Two  or  More  Fractions,  163. 


Corollary:  c  -i-  —  =  c  X  -,  162.— To  cancel  the  Denominators 
b  a 


X  TABLE  OF  CONTENTS. 

CHAPTER  XV. 

CLEARING  EQUATIONS  OF  FRACTIONS. 

Three  Classes  of  Equations  involving  Fractions,  166. 

CHAPTER  XVI. 

RADICALS  AND  SURDS. 

Rational  and  Irrational  Numbers — Radicals— Surds,  176. — Imaginary 
Quantities — Rational  Quantities  expressed  as  Radicals — Orders 

of  Radicals,  177.— Arithmetical  Roots,  178. — Theorem  :  4/^^  x 
y'b  =  ^ab,  179.  —  Reduction  of  Radicals  —  Pure  and  Mixed 
Surds.  180.  —  Theorem  :  '\/a  -r-  \^h  =  \/a-^h,  181.  —  Similar 
Quadratic  Surds — Theorem  :  m  \/a  X  n  \/a  =  mna — Theorem  : 
The  Product  of  Two  Dissimilar  Quadratic  Surds  cannot  he  Ra- 
tional —  Rationalizing  Factor,  182. — Reduction  of  Fractional 
Radicals  to  Integral  Radicals — Addition  and  Subtraction  of 
Radicals  of  the  Same  Order,  183. — Rule  for  Addition  of  Radicals 
— Rule  for  Subtraction  of  Radicals — Addition  and  Subtraction  of 
Radicals  of  Different  Orders,  184. — Multiplication  of  Radicals  of 
the  Same  Order,  185.— Simple,  Compound,  and  Conjugate  Radi. 
cals,  187. — Rationalization  of  Polynomial  Radicals,  188.— Ra. 
tionalization  of    the  Denominator  of  a  Fraction— Division  of 

Radicals  of  the  Same  Order,  189.— Theorem  :  (  1/aY  =  l/'a"  , 

191. — Theorem  :  V  \^a  =  ^j/a — To  change  Radicals  from  One 
Order  to  Another,  193. — Multiplication  and  Division  of  Radicals 
of  Different  Orders,  193. — Radical  Equations,  194. — Reduction  of 
Radical  Equations  by  Rationalization,  196. 

CHAPTER   XVII. 
THE    INDEX    LAW. 

Meaning  of  Fractional  Exponents,  198. — Meaning  of  Zero  Exponent 
— Meaning  of  Negative  Exponents,  200. — The  Index  Law  holds 
for  all  Rational  Values  of  m  and  n,  201. 

CHAPTER  XVIII. 

ELIMINATION 

Simultaneous  and  Independent  Equation,  207.  —  Two  Unknown 
Quantities  require  Two  Independent  Equations  for  their  Solution, 
208. — Elimination — Three  Methods  of  Elimination,  209. — n  In- 
dependent Equations  are  required  to  solve  for  n  Unknown 
Quantities,  216. 

CHAPTER  XIX. 

QUADRATIC  EQUATIONS  OF  ONE  UNKNOWN  QUANTITY. 

Trinomial  and  Binomial  Quadratics — Factors  of  a;^  _|_  ^^  222. — Factors 
of  a  Trinomial  Quadratic,  224. — Quadratic  Equation  of  One  tJn- 


TABLE  OF  CONTENTS.  xi 

known  Quantity  —  Roots  of  an  Equation,  237. — Solution  of  a 
Quadratic  Equation,  228.— Formation  of  Quadratic  Equations, 
280.  —  Interpretation  of  Solutions,  234.— Solution  u:t:^ -\- bx -\- 
c  =  0,  238. — Solution  of  Equations  which  are  Quadratic  in  Form, 
243. 

CHAPTER  XX. 

QUADRATIC  EQUATIONS  OF  TWO   UNKNOWN  QUANTITIES. 

Special  Cases  of  Elimination,  246. 

CHAPTER  XXI. 

INDETERMINATE  EQUATIONS  OF  THE  FIRST  DEGREE. 

Indeterminate  Equations— Solution  of  Indeterminate  Equations  of 
the  First  Degree  in  x  and  y,  259. — Solution  of  Indeterminate 
Equations  of  the  First  Degree  in  x,  y,  and  2,  263. 

CHAPTER   XXII. 

INEQUALITIES. 

Definition  of  Greater  and  Less  Quantities — Inequalities,  267. — Ele- 
mentary Theorems,  268. — Type  Forms,  273. 

CHAPTER   XXIII. 

RATIO  AND  PROPORTION. 

Definition  of  Ratio — Expression  of  Ratio,  276. — Terms  of  a  Ratio — 

Kinds  of  Ratios — Ratio  of  Equimultiples  and  Submultiples,  277. 

,„.  a-\-x        a  .     a  —  X  ^   a        .  ^    ,         . 

—  Iheorem  :  ; — ; <  ,-,    and     7* >  -,    when   a  >  b  and 

b-\-x        b  b  —  X      b 

7     rni  a  4-  X       a  .     a  —  X       a        . 

X  <  b — Theorem  :  -; — -—  >  — ,     and <  — ,    when    a  <  b 

b.-j-x       b  b  —  X       b 

and  X  <  b,  278. — Compound  Ratios — ^Definition  of  Proportion  — 
Test  of  the  Equality  of  Two  Ratios,  279. — Permutations  of  Pro- 
portions, 280. — Transformation  of  Proportions,  281. — Solution  of 
Fractional  Equations,  28-5.  —  Direct  Variation,  288.  —  Inverse 
Variation,  289. — Constant  of  Variation,  290. 

CHAPTER   XXIV. 

LOGARITHMS. 

Definition  of  a  Logarithm — Working  Rules  of  Logarithms,  293. — 
Systems  of  Logarithms — Common  Logarithms,  295. — Character- 
istic and  Mantissa,  296. — Logarithmic  Tables,  297. — Method  of 
using  Logarithmic  Tables,  299. — Cologarithms,  303. — Multiplica- 
tion by  Logarithms — Division  by  Logarithms,  304  — Involution 
by  Logarithms — Evolution  by  Logarithms,  305: — Theorem  : 
logi,m  =  \oga?n  .  logfort,  307. 


Xii  TABLE  OF  CONTENTS. 

PAET   II. 

ELEMENTARY    SERIES. 

CHAPTER   XXV. 

VARIABLES  AND  LIMITS. 

Constants  and   Variables — Functions,   311.— Limit  of  a  Variable — 
Axioms — Theorem  :    If  c  denote   any  finite   quantity,   then,  by 

taking  x  great  enough,   —  <  c — Theorem  :  If  c  denote  any  finite 

quantity,  then  by  taking  x  small  enough,  -  >  c,   312. — Infinites 

— Infinitesimals,  313.— Approach  to  a  Limit,  314. — Theorem  :  If 
k  be  any  fixed  quantity  and  s  denote  a  quantity  as  small  as  you 
please,  then,  by  taking  x  small  enough,  kx  <  s — Theorem  :  Two 
equal  functions  must  have  the  same  limit,  315. — Theorem  :  The 
limit  of  the  sum  of  several  variables  is  the  sum  of  their  limits, 
316. — Theorem  :  The  limit  of  the  product  of  two  functions  is 
the  product  of  their  limits,  317. — Theorem  :  The  limit  of  the 
quotient  of  two  functions  is  the  quotient  of  their  limits — Defini- 


tion 


of  f[-n=a  and — Theorem:  Lim.  

x-a  _\^^  x-aj 


=  w«"~i  for  all  values  of  n,  318. — Definition  of  Series — Theorem: 
The  limit  of  Ao -\- AiX-\- A.x'' +  Asx"^  .  .  .  =  Ao ,  320.— 
Theorem  :  In  the  series  Ao  +  A^x'  -\-  A-iX^  -f  .  .  .  ,  by  taking  x 
small  enough  we  may  make  any  term  as  large  as  we  please  com- 
pared with  the  sum  of  all  the  terms  which  follow  it,  and  by 
taking  x  large  enough  ^e  can  make  any  term  as  large  as  we 
please  compared  with  the  sum  of  all  the  terms  that  precede  it, 
321. — Vanishing  Fraction,  322. — Discussion  of  Problems,  324. 


CHAPTER  XXVI. 

THE  PROGRESSIONS. 

A       ARITHMETICAL   PROGRESSION. 

Arithmetical  Series — The  nth.  Term,  333.  —  Problem  :  To  find  the 
common  difference,  and  any  other  term  when  two  terms  are 
given — Arithmetical  Means,  334. — Problem  :  To  find  the  sum  of 
w  terms  of  an  A.  P.,  337.— The  Average  Term,  340. — Two  Allied 
Series,  342. 

B.      GEOMETRICAL   PROGRESSION. 

Geometrical  Series,  844. — Type  Form  of  the  Series  —  Geometrical 
Means,  345.— Problem  :  To  find  the  sum  of  n  terms  of  a  geo- 
metrical series — Divergent  and  Convergent  Series,  347. — Value 
of  Repeating  Decimals — Values  of  Recurring  Decimals,  350. 


TABLE  OF  CONTENTS.  Xlll 


C.     COMPOUND    INTEKEST   AND   ANNUITIES. 

Compound  Interest-— Problem  :  To  find  the  amount  at  compound  in- 
terest, 851. — Present  Worth,  at  Compound  Interest— Problem  : 
To  find  present  worth,  at  compound  interest,  353. — Problem  : 
To  find  the  amount  at  compound  interest  of  a  fixed  sum  invested 
at  stated  intervals,  353. — Annuities — Problem  :  To  find  the  pres- 
ent value  of  an  annuity,  354. — Problem  :  To  find  the  amount  of 
an  annuity  purchasable  at  a  given  sum — Problem  :  To  find  the 
amount  of  an  annuity  to  begin  after  m  years  purchasable  for 
a  given  sum,  355, 

D.      HARMONIC  PROGRESSION. 

Harmonic  Series— Theorem  :  If  three  quantities  are  in  harmonic 
progression,  their  reciprocals  are  in  A.  P.,  357. —  Harmonic 
Mean — Theorem  :  The  geometric  mean  of  two  quantities  is 
the  geometric  mean  of  the  arithmetic  and  harmonic  means  of 
the  quantities — Problem  :  To  insert  n  harmonic  means  between 
a  and  6,  358. 

CHAPTER  XXVII. 

BINOMIAL   THEOREM. 

Binomial  Formula,  360. — Binomial  Coefficients,  362. — Recurrence  of 
the  Coefficients — Exponent — Signs,  364. — Practical  Rules,  365. — 
General  Term,  366. — Binomial  Theorem  for  any  Rational  Index, 
368. 

CHAPTER   XXVIII. 

PERMUTATIONS  AND  COMBINATIONS. 

Permutation  —  Combination,  369.  —  Symbols  of  Combination  and 
Permutation,  370. — Number  of  Permutations,  371. — Problem  : 
To  find  the  number  of  permutations  of  7i  dissimilar  things  r  at  a 
time,  372. — Problem  :  To  find  how  many  of  the  permutations 
"Pr  contain  a  particular  object,  373 —Problem  :  To  find  the 
number  of  permutations  of  n  things  all  together  when  u  of  the 
things  are  alike,  375. — Problem  :  To  find  the  value  of  "Cr— 
Problem  :  To  find  the  number  of  times  a  particular  object  will 
be  present  in  the  combinations  "^Cr ,  376, — Meaning  of  the  Bi- 
nomial Coefficients,  378. 

CHAPTER   XXIX. 

DEPRESSION  OF  EQUATIONS. 

General  Equation  of  the  nth  Degree  in  x — Theorem  :  If  a  is  a  root  of 
the  equation  x^  -j-aiX^-^-\-  a^x^^-^-\-  .  .  .  an'-ix4-  ««  =  0,  the 
first  member  is  divisible  by  x  —  a,  381. — Converse  of  the  Theorem 
— An  Equation  of  the  nth.  Degree  has  n  Roots,  382. 


XIV  TABLE  OF  CONTENTS. 

CHAPTER  XXX. 

UNDETERMINED  COEFFICIENTS. 

Theorem :  An  integral  expression  of  tlie  nih.  degree  in  x  cannot  vanish 
for  more  than  n  vahies  of  x,  except  the  coefficient  of  all  the 
powers  of  x  are  zero — Theorem  :  If  Aic"  -\-  Bx'^~  i  -f- .  .  .  =  A'x^ 
-\-  B'X*^-^  +  •  •  •  for  all  values  of  a*,  both  functions  being  of  finite 
dimensions,  then  A  =  A',  B  =  B' ,  etc.,  386 — Definition  of  par- 
tial fractions,  387. —  Separation  of  a  fraction  into  its  partials, 
388.— Theorem  :  If  Ax^'  -\-  Bx»-^  -\- .  .  .  =  A'x^  +  J5'a;«-i  -}-..., 
for  all  values  of  x  which  make  the  series  convergent,  both  func- 
tions being  of  infinite  dimensions,  then  A  =  A',  B  =  B',  etc., 
393. — Expansion  of  Functions,  393. 

CHAPTER   XXXI. 
CONTINUED  FRACTIONS. 
Definition  of  a  Continued  Fraction— The  Convergent  of  a  Continued 
Fraction,  399.— Theorem:  ^-^  =   ^'''Pr-i  +  Vr-2  ^    ^^^   _  ^^^^:^^^ 
qr        arqr-i-i-gr-2 

and  Complete  Quotients— Theorem:  ^  -  ^^^  =  tlU!!      402. 

Qn         Qu-l  gnQn-l 

Theorem:  Each  convergent  is  nearer  in  value  to  the  continued 
fraction  than    any    previous  convergent,   404. — Theorem:    The 

V  1 

value  of  a  continued  fraction  difEers  from  — ^  by  less  than  — r 

qn  qn' 

and  by  more  than  r-^ Theorem:    The  last  convergent   pre- 

ceding  a  large  partial  quotient  is  a  close  approximation  to  the 
value  of  a  continued  fraction,  405. — Theorem:  Every  fraction 
whose  numerator  and  denominator  are  positive  integers  can  be 
converted  into  a  terminating  continued  fraction,  406. — Periodic 
Continued  Fractions — Theorem  :  A  quadratic  surd  can  be  ex- 
pressed as  an  infinite  periodic  continued  fraction,  410. — 
Theorem:  An  infinite  periodic  fraction  may  be  expressed  as  a 
quadratic  surd,  412. 


PART  I 

FUNDAMENTAL   PRINCIPLES  AND 
OPERATIONS 


ELEMENTARY  ALGEBRA. 

CHAPTER  I. 
ALGEBRAIC  NOTATION  AND  SYMBOLS. 

1.  Symbols  of  Operation. — Algebra  treats  of  the  prop- 
erties and  relations  of  numbers.  In  this  respect  algebra 
agrees  with  arithmetic. 

The  fundamental  operations  of  algebra  are  the  same  as 
those  of  arithmetic.  These  are  addition,  subtraction, 
multiplication,  division,  involution,  and  evolution. 

These  operations  are  also  indicated  by  the  same  signs  in 
algebra  as  in  arithmetic.  These  are  -|-  (plus)  for  addition, 
—  (minus)  for  subtraction,  X  for  multiplication,  -^  for 
division,  a  figure  placed  above  at  the  right  (called  an  ex- 
ponent) for  involution,  and  |/  (radical)  for  evolution. 
These  are  called  operative  symbols,  or  symbols  of  operation. 

Multiplication  is  also  indicated  by  a  dot  between  the 
factors.     Thus,  4 .  5  means  that  4  is  to  be  multiplied  by  5. 

2.  Algebraic  Expressions.  —  Numbers  are  denoted  in 
algebra  by  letters  as  well  as  by  figures.  This  is  one  respect 
in  which  algebra  differs  from  arithmetic. 

When  figures  are  written  one  after  another  in  arith- 
metic, the  expression  denotes  the  sum  of  the  different 
orders  of  units  denoted  by  the  figures  separately.  Thus, 
334  =  300  +  20  +  4. 


2  ALOEBBAIG  NOTATION  AND  SYMBOLS. 

When  letters  are  written  one  after  another  in  algebra, 
the  expression  formed  denotes  the  product  of  the  numbers 
denoted  by  the  individual  letters.     Thus,  abc  —  axh  X  c. 

When  figures  are  used  in  algebra,  they  are  combined 
to  form  numbers  in  the  same  way  as  in  arithmetic. 

When  figures  and  letters  are  written  one  after  another, 
the  expression  denotes  a  product  of  which  the  numeral  and 
literal  parts  are  factors.     Thus,  12^c  =  12  x  ^  X  c. 

Literal  expressions  are  more  comprehensive  than  nu- 
meral expressions.  Thus,  324  means  one  number  only, 
while  al)c  represents  every  product  that  is  composed 
of  three  factors,  and  these  factors  may  be  integral, 
fractional,  or  surd.  Owing  to  this  comprehensiveness 
of  its  expressions,  algebra  is  sometimes  called  generalized 
arithmetic. 

To  find  the  value  of  an  algebraic  expression  is  to  find 
the  number  which  it  represents  on  the  supposition  that  its 
letters  stand  for  particular  numbers. 

3.  Exponents. — When  the  same  letter  enters  more  than 
once  as  a  factor  in  a  product,  the  number  of  times  that  it 
enters  as  a  factor  is  indicated  by  writing  a  figure  after  it 
at  the  top.  Thus,  aU^c^  —  a  X  h  X  i  X  c  X  c  X  c.  The 
expression  is  read  ''  a,  h  square,  c  cube,"  or  ^'  a,  l  second, 
c  third." 

The  number  used  to  denote  how  many  times  the  same 
factor  occurs  in  a  product  is  called  an  exponent. 

4.  Coefficients. — The  number  used  to  denote  how  many 
times  a  single  letter  or  a  product  of  two  or  more  letters  is 
taken  is  written  before  the  letter  or  product  and  on  a  line 
with  it. 

The  number  thus  used  is  called  a  coefficient.  Thus,  6x 
denotes  that  the  number  x  is  taken  5  times.  That  is,  6x  = 
x-]-x-]-x-]-x-\-x;  while  x^  {x fifth)  =  x  X  x  X  x  X  x  X  x. 
7abc  =  abc  +  abc  -{-  ahc  +  ahc  +  ahc  +  aic  +  aho. 


ALGEBRAIC  NOTATION  AND  SYMBOLS.  3 

When    no    coefficient    or   exponent  is  expressed,    the 
number  one  is  to  be  assumed. 

EXERCISE    I. 

Find  the  value  of  the  following  expressions  when  a  =  d, 

h  —  b,  and  c  =  7 : 


1.     abc. 

2.     5abc.                   3.     ab^c. 

4.     4.a'bc\ 

5.     Gd'b^c.                6.     12a3^,V. 

7.     2oab''^c^. 

8.     4:0a^Pc.              9.     75aHc^ 

10.     250a^h. 

11.     Find  the  cost  of  a  oranges  at  5  cents  a  piece. 

12.     Find  the  surface  of  a  rectangular  board  10  ft.  long 

and  a  inches  wide. 

13.  There  are  twenty  pages  in  a  book,  and  on  each  page 
there  are  m  lines,  and  in  each  line  n  words.  How  many- 
words  in  the  book  ? 

14.  There  are  a  drawers  in  a  case,  a  compartments  in 
each  drawer  and  c  specimens  in  each  compartment,  and 
there  are  25  cases  in  a  room       How  many  specimens  in  all 

the  cases  ? 

5.  Numeric  Values. — A  magnitude  is  any  thing  which 
has  size  or  extent,  and  which  is  doubled  when  added  to 
itself.     Thus,  lengths  and  distances  are  magnitudes. 

Magnitudes  are  measured  by  comparing  them  with 
some  other  magnitude  of  the  same  kind,  to  see  how  many 
times  they  contain  it. 

The  magnitude  with  which  other  magnitudes  are  com- 
pared in  measurement  is  called  the  unit  of  measure^nent^ 
or  the  iniit  magnitude. 

When  the  magnitude  contains  the  unit  an  exact  number 
of  times  the  number  which  expresses  how  many  times  a 


4  ALGEBRAIC  NOTATION  AND  SYMBOLS. 

magnitude  contains  the  unit  is  called  the  numeric  value  of 
the  magnitude.  This  term  is  also  extended  to  the  cases  in 
which  the  value  can  he  expressed  only  by  a  fraction  or  a 
surd. 

Numerical  expressions,  whether  composed  of  figures  or 
letters  or  of  both,  are  called  quantities.  Every  algebraic 
expression  is  numerical;  that  is,  it  represents  some  num- 
ber.    Hence  every  algebraic  expression  is  a  quantity. 

6.  Quantitative  Symbols. — The  symbols  which  express 
number  are  called  quantitative  symbols.  In  algebra  they 
are  both  numeral  and  literal. 

7.  Terms. — When  an  algebraic  expression  is  made  up 
of  parts  separated  by  signs  of  operation,  the  parts  separated 
by  the  consecutive  signs  are  called  terms. 

Thus,  in  the  expression  bci^b  +  c  —  12  -f  ati^c,  6aH,  c, 
12,  and  alP'c  are  terms. 

It  will  be  noticed  that  a  term  may  be  a  single  letter,  a 
number  expressed  by  one  or  more  figures,  or  a  product  com- 
posed of  literal  or  of  literal  and  numeral  factors.  The  nu- 
meral factor  of  a  term  is  commonly  called  its  coefficient,  and 
when  no  numeral  factor  is  expressed  the  coefficient  is  to  be 
regarded  as  one. 

Thus,  in  the  expression  7a^?/  —  5«  +  ^•^^.'  the  coefficient 
of  the  first  term  is  7,  of  the  second  5,  of  the  third  1. 

8.  Monomials  and  Polynomials. — An  algebraic  expres- 
sion which  contains  no  signs  of  operation  is  called  a  mo- 
nomial, or  a  one-term  expression;  one  composed  of  two 
terms  separated  by  a  sign  of  addition  or  subtraction,  a  hi- 
nomial,  or  a  two- term  expression ;  one  composed  of  three 
terms  separated  by  signs  of  addition  or  subtraction,  a  tri- 
nomial, or  a  three-term  expression.  Expressions  which 
contain  more  than  three  terms  are  sometimes  called  multi- 


ALGEBRAIC  NOTATION  AND  SYMBOLS.  5 

nomials,  and  all  expressions  which  contain  more  than  one 
term  are  usually  classed  together  as  polynomials. 

To  find  the  value  of  a  polynomial,  we  must  find  the 
value  of  each  of  its  terms  and  then  add  or  subtract  these 
values  according  to  the  signs  before  the  terms.  Every 
minus  term  of  a  polynomial  must  be  subtracted  from  the 
sum  of  the  plus  terms  or  from  some  individual  plus  term. 

AVhen  no  sign  is  placed  before  the  first  term  of  a  poly- 
nomial it  is  understood  to  be  a  plus  term. 

EXERCISE  II. 

Find  the  value  of  each  of  the  following  polynomials, 
when  «  =  3,  ^  =  1/2,  and  c  =  2/3 : 

1.  5  4-  a^c  -  2abc  -  Wc  +  lOa^c^. 

2.  9ac^  -  24:bh  -  (Jab^  -  ISabc^  +  7a^c. 

Find  the  values  of  the  following  polynomials  when 
a  =  2,  b  =  d,  c  =  4:,  and  d  —  b: 

3.  Qabcd  —  5«^6'  —  7«^^^+  'da^cd^* 

5.  ^a^cd^  —  lab'^d  -\-  iSabcd  —  6d^c. 

6.  -  5a^c  +  Sa^cd'  +  (Jabcd  -  lab^d. 

Note  that  the  value  of  a  polynomial  remi^ins  the  same 
in  whatever  order  its  terms  are  written. 

Note  also  that  the  value  of  a  polynomial  may  be  found 
by  first  adding  together  the  values  of  its  plus  terms,  and 
also  of  its  minus  terms,  and  then  subtracting  the  latter 
sum  from  the  former. 

9.  Similar  Terms.  —  Similar  terms  are  those  which 
agree  both  in  their  letters  and  in  their  exponents.  They 
need  not,  however,  agree  either  in  their  signs  or  in  their 
coefficients.  Thus  a^xy^,  ba^xy^,  —  Zc^xy^,  are  all  similar 
terms. 


6  ALGEBRAIC  NOTATION  AND  SYMBOLS. 

The  similar  terms  of  a  polynomial  may  be  combined 
into  one  term  by  performing  upon  their  coefficients  the 
operations  indicated  by  the  signs  of  the  term,  and  using 
the  resulting  number  as  the  coefficient  of  the  common 
literal  factors  of  the  terms. 

Dissimilar  terms  cannot  thus  be  combined  into  one. 

Similar  plus  terms  are  combined  into  one  plus  term  by 
adding  their  coefficients,  similar  minus  terms  are  combined 
into  one  minus  term  by  adding  their  coefficients,  and  a  plus 
and  a  minus  term,  when  similar,  are  combined  into  one  by 
subtracting  their  coefficients. 

EXERCISE  III. 

Eeduce  the  following  polynomials  to  simpler  forms  by 
combining  their  similar  terms : 

1.  9a^^  +  lOa^^  -  ^a%^  -  ^aW  +  12. 

2.  12«  -  5^2  _  6^  _  7^2  _  2«  _  3^  _^  6  -  3. 

3.  -  Qxhj  +  8  -  ^x^y  +  Ibx'y  -  10  +  7  -  5Z»3. 

4.  7«2^  -  l^ahj  +  ^ay^  +  'da'y  -  a^y  -  7. 

6.  -  7 A  +  12A  -  5A3  _  Qa^x  +  %a^x  +  15-9. 


CHAPTER  II. 
EQUATIONS  AND  PARENTHESES. 

A.    EQUATIONS. 

"10.  Members  of  an  Equation. — An  algebraic  expression 
of  equality  is  called  an  equation.  It  is  composed  of  two 
members  separated  by  the  sign  of  equality.  The  part  be- 
fore the  sign  of  equality  is  called  the/rs^  member,  and  the 
part  after  the  sign,  the  second  member. 

Thus,  7x  —  2^;  +  6  =:  26  +  ^  is  an  equation.  Ix  —  2x 
-[-  6  is  its  first  member,  and  26  +  2;  is  its  second  member. 

11.  Verbal  Symbols. — The  signs  =,  >,  <,  .'.  stand 
for  the  phrases  ''equal  to,"  "greater  than,"  "less  than," 
"  therefore"  or  "then,"  and  are  hence  called  verbal  signs. 

12.  Axioms. — A  mathematical  truth  so  evident  as  to  be 
generally  accepted  without  proof  is  called  an  axmn.  The 
following  are  important  axioms  about  equations. 

1°.  If  the  same  quantity  or  equal  quantities  be  added 
to  equals,  the  sums  will  be  equal. 

2°.  If  the  same  quantity  or  equal  quantities  be  sub- 
tracted from  equals,  the  remainders  will  be  equal. 

3°.  If  equals  be  multiplied  by  the  same  quantity  or  by 
equal  quantities,  the  products  will  be  equal. 

4°.  If  equals  be  divided  by  the  same  quantity  or  by 
equal  quantities,  the  quotients  will  be  equal. 

5°.  The  same  powers  of  equals  are  equal. 

6°.  The  same  roots  of  equal  quantities  are  equal. 

7 


8  EQUATIONS  AND  PARENTHESES. 

The  two  following  axioms  are  applicable  to  all  algebraic 
expressions. 

7°.  The  subtraction  of  any  quantity  from  an  algebraic 
expression  neutralizes  the  effect  of  its  addition  to  the  ex- 
pression. 

8°.  The  division  of  an  algebraic  expression  by  any 
quantity  neutralizes  the  effect  of  multiplying  the  expression 
by  the  same  quantity. 

13.  Transposition  of  Terms.  —  It  follows  from  axioms 
1°  and  2°  that  a  term  may  be  omitted  from  one  member  of 
an  equation  and  written  with  the  opposite  sign  in  the  other 
without  destroying  the  equality  of  the  members. 

Thus,  if  7a;  -  2a;  +  6  =  26  +  a;,  then,  by  axiom  2°, 

7x  —  2x  —  X  -\-  6  —  26  -{-  X  —  X,  and,  by  ax.  7°, 
7x  —  2x  —  X  -{-  Q  =  2Q.  Again,  by  axiom  2°, 
7a;  -  2a;  -  a;  +  6  -  6  =  26  -  6,  and,  by  ax.7°, 

7a;  -  2a;  -  a;  =  26  -  6. 

When  a  term  is  omitted  in  one  member  and  placed  with 
the  opposite  sign  in  the  other  it  is  said  to  be  transposed. 
A  plus  term  is  transposed  by  subtracting  it  from  each 
member,  and  a  minus  term  by  adding  it  to  each  member. 

Combining  the  similar  terms  in  the  last  equation,  we  get 

4a;  =  20. 

14.  Collection  of  Terms. — The  combining  of  the  similar 
terms  in  an  equation  is  called  collecting  the  terms. 

15.  Division  by  the  Coefficient  of  x. — Dividing  each 
member  of  the  equation  4a;  =  20  by  4  we  get,  by  axiom  4°, 
X  =  5. 

16.  Solution  of  an  Equation. — To  solve  an  equation  is 
to  find  the  value  in  terms  of  known  quantities  of  the  letter 
in  it  which  represents  an  unknown  quantity. 


EQUATIONS.  9 

It  is  customary  to  represent  known  quantities  by  the 
first  letters  of  the  alphabet  and  unknown  quantities  by  the 
last  letters,  x,  y,  z,  etc. 

Among  the  steps  necessary  to  the  solution  of  an  equa- 
tion are  transposition,  collection,  and  division  by  the  coef- 
ficient of  the  unknown  quantity. 

EXERCISE   IV. 

Solve  each  of  the  following  equations,  and  name  and 
explain  each  step  taken: 

1.  ^x  +  ^x  -  12  =  bx  +  72. 

2.  14^  +  8  —  "ly  =99  —  2/, 

3.  8;^  -  5  +  6  +  2^  =  3^  +  53. 

4.  l/2x  +  3/2:?;  -  .T  +  7  =  27  -  l/3a;. 

5.  7/52;  -  l/3x  -  18  =  72  +  d/4:X, 

6.  Sx  -\-  a  =  b  -\-  5a. 

7.  ax  -{-  d  -\-  3ax  —  c  —  hax. 

17.  Literal  Coefficients. — In  the  seventh  example,  a 
may  be  considered  as  the  coefficient  of  x  in  the  first  term, 
3«  as  the  coefficient  of  x  in  the  third  term,  and  5«  as  the 
coefficient  of  x  in  the  last  term.  Coefficient  means  felloiu 
factor,  and  in  any  literal  product  all  the  factors  but  one 
may  be  taken  as  the  coefficient  of  that  factor. 

18.  Algebraic  Solution  of  Problems. — To  solve  a  prob- 
lem algebraically,  wo  must  first  obtain  an  equation  in  terms 
of  the  known  and  unknown  quantities  of  the  problem,  and 
then  solve  the  equation  to  find  the  value  of  the  unknown 
quantities  in  terms  of  the  known. 

e.g.  1.  Divide  the  number  105  into  two  parts,  one  of 
which  shall  be  six  times  the  other. 

Let  X  =  the  number  in  the  smaller  part; 

.  •,   6x  =  the  number  in  the  larger  part. 


10  EQUATIONS  AND  PARENTHESES. 


and 

Qx-{-  X  —  the  whole  number. 

Also 

105  =  the  whole  number; 

.  • 

.  Qx^x^  105. 

Collfccting, 

rx  =  105. 

Dividing  by  7, 

a;  =  15. 

.• 

.   Qx  =  90. 

The  numbers  are  15  and  90. 

e.g.  2.  Eight  times  the  smaller  of  two  numbers  is 
equal  to  143  minus  the  larger,  and  the  larger  is  three  times 
the  smaller.     Find  the  numbers. 

Let  X  =  the  smaller  number ; 

.  •.  3x  =  the  larger  number. 

.-.   Sx=  143  -dx. 

Transposing,         8x  -\-  dx  =  143. 

Collecting,  llo;  =  143. 

Dividing  by  11,  a;  =    13. 

.-.  3x=    39. 

The  numbers  are  13  and  39. 

EXERCISE  V. 

I. 

1.  rind  two  numbers  whose  difference  is  9  and  whose 
sum  is  63. 

2.  Divide  103  into  two  parts  whose  difference  shall  be 
13. 

3.  Find  two  numbers  such  that  the  larger  shall  be  4 
times  the  smaller,  and  that  6  times  the  smaller  shall  equal 
60  plus  the  larger. 

4.  Find  two  numbers  such  that  the  larger  shall  be  5 
times  the  smaller,  and  that  7  times  the  smaller  shall  equal 
374  minus  3  times  the  larger. 

5.  Divide  450  into  three  .parts  such  that  the  second  shall 


EQUATIONS.  11 

contain  twice  as  many  as  the  third,  and  tlie  first  tliree  times 
as  many  as  the  third. 

II. 

6.  120  marbles  are  arranged  in  3  piles  so  that  there  are 
twice  as  many  marbles  in  the  first  pile  as  in  the  second  and 
tliree  times  as  many  in  the  second  as  in  the  third.  How 
many  marbles  in  each  pile  ? 

7.  In  a  scliool  there  are  three  grades,  and  there  are  three 
times  as  many  scholars  in  the  lowest  grade  as  in  the  middle 
grade  and  five  times  as  many  in  the  middle  grade  as  in  the 
highest.  The  whole  school  numbers  735.  How  many 
scholars  are  there  in  each  grade  ? 

8.  A  man  bought  a  horse,  a  carriage,  and  a  harness  for 
450  dollars.  He  paid  three  times  as  much  for  the  horse  as 
for  the  harness,  and  twice  as  much  for  the  carriage  as  for 
the  horse.     What  was  the  cost  of  each  ? 

9.  A  boy  bought  a  speller,  an  arithmetic,  and  a  history 
for  $2.30.  He  gave  twice  as  much  for  the  history  as  for 
the  arithmetic,  and  three  times  as  much  for  the  arithmetic 
as  for  the  speller.     How  much  did  he  pay  for  each  ? 

10.  A  boy  is  three  years  older  than  his  sister,  and  has  a 
brother  who  is  five  years  older  than  himself.  Their  united 
ages  are  41  years.     How  old  is  he  ? 

19.  Clearing  Equations  of  Fractions. — Since  a  fraction 
is  reduced  to  its  numerator  when  it  is  multiplied  by  its  de- 
nominator, and  since  botli  members  of  an  equation  may  be 
multiplied  by  the  same  number  without  destioying  their 
equality,  an  equation  may  be  freed  of  a  fraction  by  multi- 
plying both  its  members  by  the  denominator  of  the  frac- 
tion. 

Zx 
e.g.  Free  the  equation  — -  =  6  of  its  fraction. 

0 


12  EQUATIONS  AND  PARENTHESES. 

^X5  =  6X5.     (Why?) 

0 

.'.      ^x  =  30; 
a;  =  10. 

Note  that  8  +  4  multiplied  by  2  =  either  12  X  2  =  24 
or  8  X  2  +  4  X  2  =  16  +  8  =  24.  Also  that  8-4  mul- 
tiplied by  2  =  either  4x2  =  8  or  8x2-4x2  =  10 
-8  =  8. 

So  in  general  a  -\-  h  multiplied  oy  2  =  2a  +  2b,  and 
o,  —  b  multiplied  by  2  =  2a  —  2b.  That  is,  to  multiply  any 
algebraic  expression  by  a  number,  we  must  multiply  each 
term  of  the  expression  by  the  number. 

If  an  equation  contains  two  or  more  fractions  it  may  be 

freed  of  all  of  them  by  multiplying  both  its  members  by 

the  product  of  all  the  denominators  at  once. 

2x        ^'x 
e.g.  Free  the  equation  — -  -|-  — —  =  8  of  fractions. 

Multiplying  both  members  by  12,  we  get 
^+^-96 

or  Sx  +  9a:  =  96. 

Instead  of  multiplying  both  members  by  the  product  of 

all  tlie  denominators,  we  may  multiply  by  the  least  common 

multiple  of  the  denominators. 

2x       3^/       A:X 
e.g.  Free  the  equation  -;j-  +  t — H  To"  ~  ^  ^^  fractions, 
o  o         12 

The  L.  C.  M.  of  3,  5,  and  12  is  60.     Multiplying  both 

members  by  this,  we  obtain 

120a;   ,   180a;   ,   240aj  _ 
or        40a;  +  36a;  +  20a;  =  120. 


EQUATIONS.  13 

Ex.   1.  Divide  150  into  two  parts  such  that  the  first  shall 
be  2/3  of  the  second. 

Let 


Hence 


or 


X 

= 

the  number 

in 

the  second 

part; 

2x 
3 

= 

a 

a 

<i 

"   first 

(( 

X 

+ 

2x 
3 

--  150. 

'Sx 

+ 

2x  = 

450, 

5x 

= 

450. 

X 

= 

90. 

2x 

= 

60. 

Hence  the  parts  are  GO  and  90. 

Ex.  2.  Divide  $37.20  among  four  men  so  that  the 
second  shall  have  2/3  as  much  as  the  first,  the  third  3/4  as 
much  as  the  second,  and  the  fourtli  5/6  as  much"  as  the 
third. 

Let  X   —  number  of  dollars  received  by  the  first, 

2a; 
.•.-—=        "        "      ''  "         ''     ''   second, 

o 

:^-  =1=        "      "  ''         "     *^   third, 

30it'       hx 
and        -^^  =:  _  zr:    '*      ^-  '^         "     '^  fourth. 

7/4  1/i 

.%  12a;  +  8a;  4-  6./;  +  5a;  =  446.40, 
0-.  31a;  ==  446.40, 
.%       a;=:    14.40. 


14  EQUATIONS  AND  PARENTHESES. 

-^  =  9.60, 
o 

1  =  7.20, 

and         If  =  6.00. 

Hence  the  first  receives  $14.40,  the  second  19.60,  the 
third  $7.20,  and  the  fourth  $6.00. 

EXERCISE  VI. 

1.  Divide  175  into  two  parts,  so  that  the  first  shall  be 
2/3  of  the  second. 

2.  Two  men  in  comparing  their  ages  found  that  tlie  first 
was  3/5  as  old  as  the  second,  and  that  their  united  ages 
were  72  years.     How  old  was  each  ? 

3.  Divide  $4.89  among  four  boys  so  that  the  second 
shall  receive  3/2  as  much  as  the  first,  the  third  3/4  as 
much  as  the  second,  and  the  fourth  2/5  as  much  as  the 
third. 

4.  A  man  bought  four  houses  for  $117,000.00.  He 
paid  2/3  as  much  for  the  second  as  for  the  first,  4/5  as 
much  for  the  third  as  for  the  second,  and  3/4  as  much  for 
the  fourth  as  for  the  third.  How  much  did  he  pay  for 
each  ? 

5.  A  man  buys  three  horses  for  $325.00,  and  pays  four 
times  as  much  for  the  first  as  for  the  second,  and  twice  as 
much  for  the  third  as  for  the  first.  How  much  does  he  pay 
for  each  ? 

B.    PARENTHESES. 

20.  Symbols  of  Aggregation. — To  indicate  that"  any 
portion  of  an  algebraic  expression  which  lies  between  non- 


PARENTHESES.  15 

consecutive  signs  is  to  be  taken  together  as  a  complex  term, 
we  enclose  the  portion  within  parentheses  or  brackets. 

Thus,  in  the  expression  5  +  4^6'  —  3(4«  +  ^^)^  ^  and 
4«c  are  simple  terms,  3(4^  -|-  2^)  is  a  complex  term.  The 
3  may  be  considered  as  the  coefficient  of  the  parenthesis, 
and  the  minus  sign  means  that  three  times  the  quantity 
within  the  parenthesis  is  to  be  subtracted  from  what  pre- 
cedes it. 

The  parenthesis  does  not  indicate  an  operation,  but 
that  certain  parts  of  an  algebraic  expression  are  to  be  taken 
together  in  an  operation.  Hence  it  is  called  a  sign  of 
aggregation, 

A  bar  or  vinculum,  drawn  over  or  under  the  parts  of 
the  expression  which  ivq  to  be  taken  together  in  an  opera- 
tion, is  often  used  instead  of  a  parenthesis  as  a  sign  of 
aggregation. 


Thus,  5  +  4«c  -  3  .  4a  +  2^*. 

21.  Signs  of  Parenthetic  Terms. — When  two  or  more 
minus  terms  occur  in  an  expression,  they  are  to  be  sub- 
tracted from  the  remaining  terms. 

Thus,  16  —  6—4  means  that  both  the  6  and  the  4  are 
to  be  subtracted  from  16.  The  final  result  will  be  6.  This 
is  the  same  result  that  would  be  obtained  by  subtracting  10, 
the  sum  of  4  and  6,  from  16.     That  is, 

16  _  6  -  4  =  16  -  (6  +  4). 
In  general, 

a  —  h  —  c  =  a  —  {h  -\-  c). 

Again,  16  —  6  -f-  4  or  16  +  4  —  6  means  that  6 
is  to  be  subtracted  from  the  sum  of  16  and  4.  We  may 
first  take  6  from  16  and  add  4  to  the  result,  or  we  may 
first  add  4  to  the  16  and  then  take  6  from  the  result.  In 
either  case  the  final  result  will  be  14.     This  is  the  same  re- 


16  EqUATIONS  AND  PARENTHESES. 

suit  that  would  be  obtained  by  taking  the  difference  between 
4  and  6  from  16.     That  is, 

16  -  6  H- 4  =  16  -  (6  -4). 

In  general, 

a  —  1)  -\-  c  —  a  —  (1)  —  c). 

That  is,  if  a  parenthesis  have  a  minus  sign  before  it, 
the  sign  of  every  term  within  the  parenthesis  must  be 
changed  both  on  putting  on  and  on  taking  off'  the  paren- 
thesis. This  is  a  very  important  rule  and  should  be  care- 
fully borne  in  mind. 

The  expression  16  -f-  (6  —  4)  means  that  the  difference 
between  4  and  6  is  to  be  added  to  16.  The  result  is  18. 
This  is  the  same  result  that  would  be  obtained  by  first 
adding  6  to  16  and  then  taking  4  from  the  result.  That 
is, 

16  +  (6  -  4)  =  16  +  6  -  4. 

In  general, 

a-\-  {h  —  c)  —  a  -{-  b  —  c. 

That  is,  if  a  parenthesis  have  a  plus  sign  before  it,  the 
signs  of  the  terms  within  it  are  not  to  be  changed  either  on 
putting  on  or  on  taking  off  the  parenthesis. 

22.  Parenthetic  Factors. — 

4(6-4)  =  4x2  =  8==4x6-4x4. 
In  general, 

4:(b   -   C)=4:b-   4:C, 

and  4«($  —  c)  =  4«^  —  4«c. 

That  is,  in  removing  a  parenthesis,  every  term  within 
the  parenthesis  must  be  multiplied  by  the  factors  without 
the  parenthesis,  and  on  putting  on  a  parenthesis  all  fac- 


PARENTHESES.  1 7 

tors  common  to  all  the  terms  within  the  parenthesis  may  be 
placed  without  the  parenthesis. 

EXERCISE  VII. 

Kemove  the  parenthesis  from  each  of  the  following 
expressions : 

1.  da-4:b-  2a(Sb  -  4^)  +  6. 

2.  dm  +  4:n  —  5c(4:X  —  5y  -\-  g). 

3.  7  +  S{'dc  -  4:b)  -  VZx. 

4.  6x  —  a(b  -\-  c)  -{-  7a. 

.  6.     18m  +  8(2«  -  3^*  +  4c). 

6.  2x  +  d('Zx  +  7). 

Place  the  three  terms  after  the  first  of  each  of  the  fol- 
lowing expressions  within  a  parenthesis, — first  with  a  minus 
and  then  with  a  plus  sign  before  the  parenthesis  • 

7.  6x-3a-Qb  +  dc  +  9. 

8.  7ab  -  Sbc  +  IQcd  -f  Mc^  +  3. 

9.  27  +  Qa^c  -  lOa^  +  12a\ 
10.  10a:  -f  20:^2  _^  25  A  -  35. 

EXERCISE  VIII. 

I. 

1.  Find  two  numbers  whose  difference  is  4,  and  such 
that  three  times  the  less  plus  four  times  the  greater  shall 
eqtial  232  minus  eight  times  the  sum  of  the  numbers. 

2.  Find  two  numbers  whose  difference  is  6,  and  such 
that  seven  times  the  greater  minus  five  times  the  less-  shall 
equal  156  minus  nine  times  the  sum  of  the  numbers. 

3.  A  man  bought  a  carriage,  a  horse,  and  a  harness  for 
720  dollars.     He  paid  three  times  us  much  for  the  horse  as 


18  EQUATIONS  AND  PARENTHESES. 

for  the  harness,  and  twice  as  much  for  the  carriage  as  for 
the  horse  and  harness  together.  How  much  did  he  pay  for 
each  ? 

4.  A  merchant  received  131,640.00  in  three  months. 
The  second  month  he  received  80  dollars  less  than  three 
times  as  much  as  he  received  the  first  month,  and  the 
third  month  he  received  40  dollars  less  than  three  times  as 
much  as  he  received  the  first  two  months.  How  much  did 
he  receive  each  month  ? 

6.  What  number  increased  by  one-half  and  one-fifth  of 
itself  will  equal  34  ? 

II. 

6.  What  number  increased  by  two-thirds  and  three- 
fourths  of  itself,  and  21  more,  will  equal  three  times  itself? 

7.  What  number  increased  by  one-half  and  one- third 
of  itself,  and  17  more,  will  equal  50  ? 

8.  What  number  diminished  by  three-fourths  and  one- 
sixth  of  itself,  and  6  more,  will  equal  5  ? 

9.  What  number  diminished  by  two-thirds  and  one- 
ninth  of  itself,  and  11  more,  will  equal  one-ninth  of  itself? 

10.  Divide  119  into  three  parts  such  that  the  second 
shall  be  three  times  the  remainder  obtained  by  subtracting 
9  from  the  first,  and  the  third  shall  be  twice  the  remainder 
obtained  by  subtracting  the  first  from  the  second. 

23.  Note. — For  the  present  it  will  be  necessary  .to 
transpose  the  terms  of  an  equation  in  such  a  way  that, 
after  the  terms  have  been  collected,  the  term  containing 
the  unknown  quantity  will  be  plus. 

It  makes  no  difference  whether  the  unknown  quantity 
is  finally  in  the  first  or  the  second  member  of  the  equation. 

e.g.  In  a  school  of  three  grades,  one-half  the  scholars 


PARENTHESES.  19 

are  in  the  lowest  grade,  one-third  in  the  middle  grade,  and 
60  in  the  highest  grade.  How  many  scholars  in  each 
grade,  and  in  the  whole  school  ? 

Let  X  =  the  number  of  scholars  in  the  whole  school. 
.  •.     1/^x  =  the  number  of  scholars  in  the  lowest  grade, 
1/^x  —  the  number  of  scholars  in  the  middle  grade, 
and      60  =  the  number  of  scholars  in  the  highest  grade. 

.-.     1/22;  +  1/32:  +  60  =  a;, 

or  Zx-\-%x-\-  360  =  Qx, 

.  •.     360  =  &x  —  dx  —  2x, 

.'.     dQO  =  X  =  whole  school. 

1/22;  =  180;  1/32;  =  120. 

The  equation  might  have  been  written 

X  =  1/22;  +  1/32;  +  60, 

and  all  the  terms  containing  x  might  then  have  been  trans- 
ferred to  the  first  member. 

EXERCISE  IX. 


1.  A  bin  contains  a  mixture  of  rye,  barley,  and  wheat. 
2/5  of  the  grain  are  rye,  2/7  barley,  and  77  bushels  are 
wheat.  How  many  bushels  of  grain  are  there  in  all,  and 
how  many  of  each  kind  ? 

2.  In  an  orchard  there  are  three  kinds  of  apple-trees. 
2/3  of  the  trees  are  baldwins,  2/11  greenings,  and  35  are 
pippins.  How  many  trees  are  there  in  all,  and  how  many 
of  each  kind  ? 

3.  There  are  four  villages  on  a  straight  road.  The 
distance  from  the  first  to  the  second  is  3/8  of  the  distance 
from  the  first  to  the  fourth,  the  distance  from  the  second 


20  EQUATIONS  AND  PARENTHESES. 

to  the  third  is  2/5  of  that  distance,  and  the  distance  from 
the  third  to  the  fourth  is  18  miles.  How  far  are  the  vil- 
lages apart  ? 

4.  Louis  had  four  times  as  many  stamps  as  Howard, 
and  after  Louis  had  bought  80  and  Howard  had  sold  30 
they  had  together  450.     How  many  had  each  at  first  ? 

II. 

6.  Divide  226  into  three  parts,  such  that  the  first  shall 
be  four  less  than  the  second  and  nine  greater  than  the 
third. 

6.  In  an  election  70,524  votes  are  cast  for  four  candi- 
dates. The  losing  candidates  received  respectively  812, 
532,  and  756  votes  less  than  the  winning  candidate.  How 
many  votes  did  each  candidate  receive  ? 

7.  Four  towns  M,  iV,  S,  and  T  are  on  a  straight  road. 
The  distance  from  M  to  T  is  108  miles,  the  distance  from 
JVto  Sis  2/7  of  the  distance  from  M  to  JV,  and  the  dis- 
tance from  >S'  to  2"  is  three  times  the  distance  from  M  to  S. 
Find  the  distance  from  M  to  JV,  from  JV  to  S,  and  from  S 
to  T, 


CHAPTER  III. 
NEGATIVE  QUANTITIES. 

24.  Counting. — The  fundamental  relations  of  numbers 
are  determined  by. counting,  and  the  fundamental  opera- 
tions of  arithmetic  and  algebra,  when  they  are  performed 
on  integers  and  result  in  integers,  are  simply  abbreviated 
methods  of  counting. 

Numbers  may  be  counted  forward  or  backward.  In  the 
former  case  the  numbers  obtained  are  always  increasing  and 
in  the  latter  case  decreasing.  In  arithmetic  we  may  count 
forward  indefinitely,  but  backward  only  to  zero. 

Counting  forward  is  counting  on,  or  addition ;  counting 
backward  is  counting  off,  or  subtraction.  In  arithmetic 
subtraction  is  impossible  when  the  number  to  be  subtracted, 
or  counted  off,  contains  more  units  than  the  number  from 
which  it  is  to  be  subtracted,  or  counted  off.  8  —  12  rep- 
resents an  operation  which  is  arithmetically  impossible. 

In  algebra  the  operation  is  generalized,  and  counting 
off  is  considered  to  be  as  unlimited  as  counting  on.  Num- 
bers, instead  of  running  only  forward  from  zero  as  in  arith- 
metic, are  considered  as  running  backward  from  zero  as 
well. 

25.  Signs  of  duality. — In  arithmetic  the  scale  of  num- 
bers begins  at  zero  and  runs  forward  only,  while  in  algebra 
it  runs  both  ways  from  zero  at  the  centre.  To  indicate  in 
which  part  of  the  algebraic  scale  a  number  belongs,  the 
forward  part  of  the  scale  is  called  tlie  positive  part,  and  the 

31 


22  NEGATIVE  QUANTITIES. 

numbers  in  this  part  of  the  scale  are  either  written  without 
a  sign  or  are  preceded  by  a  plus  sign.  The  numbers  are 
called  positive  numbers,  and  the  plus  sign  so  used  is  called 
the  positive  sign.  The  backward  part  of  the  scale  is  called 
the  negative  part,  and  numbers  in  this  part  of  the  scale  are 
written  with  a  minus  sign  before  them.  These  numbers 
are  called  negative  numbers,  and  the  minus  sign  so  used  is 
called  the  negative  sign. 

The  signs  -|-  and  —  perform  a  double  office  in  algebra. 
They  indicate  the  operations  of  addition  and  subtraction, 
and  also  whether  a  quantity  is  to  be  taken  in  the  positive 
or  the  negative  sense.  In  the  former  case  they  are  properly 
called  plus  and  minus,  and  are  symbols  of  operation  and  in 
the  latter,  positive  and  negative,  and  are  symbols  of  quality 
or  sense.  When  a  term  stands  alone  the  sign  before  it  is 
to  be  regarded  as  positive  or  negative. 

A  term  standing  alone  without  a  sign  is  understood  to 
be  positive. 

26.  The  Algebraic  Scale  of  Numbers. — Counting  along 
the  algebraic  scale  towards  the  positive  end  is  counting  on, 
or  in  the  positive  direction,  and  counting  along  the  scale 
towards  the  negative  end  is  counting  off,  or  in  the  negative 
direction. 

The  algebraic  scale  may  be  represented  by  a  horizontal 
line  of  numbers  with  zero  at  the  centre  and  the  consecutive 
numbers  differing  by  a  single  unit,  those  to  the  right  of 
zero  being  distinguished  by  the  positive  sign,  and  those  to 
the  left  of  zero  by  the  negative  sign.     Thus, 

\       13,   12,   11,   fo,   9,   8,   7,   6,   5,  4,   3,  2,   1,  0, 

+    +    +    +    +•+    +    +   ++      +      rf      4- 
1,   2,   3,  4,   5,   6,   7,   8,   9,   10,   11,   12,   13. 

Counting  along  this  line  from  any  point  towards  the 


NEGATIVE  QUANTITIES.  23 

right  is  counting  forward,  or  positively y  and  from  any  point 
towards  the  left  is  counting  backward,  or  negatively. 

e.g.  Beginning  at  minus  five  and  counting  positively, 
we  have  minus  five,  minus  four,  minus  three,  minus  two, 
minus  one,  zero,  one,  two,  three,  four,  five,  etc.  In  this 
case  each  new  number  mentioned  is  one  greater  than  the 
last,  minus  four  being  one  greater  than  minus  five. 

Beginning  at  five  and  counting  negatively,  we  have 
five,  four,  three,  two,  one,  zero,  minus  one,  minus  two, 
minus  three,  minus  four,  minus  five,  etc.  In  this  case 
each  new  number  mentioned  is  one  less  than  the  last. 

Whatever  a  positive  unit  may  be,  the  corresponding 
negative  unit  is  something  just  the  opposite. 

27.  Absolute  and   Actual  Values  of  Numbers.  —  The 

absolute  value  of  a  number  is  the  number  of  units  in  it  ir- 
respective of  their  sign,  while  its  actual  value  is  its  value 
due  to  the  number  and  sign  of  its  unit.  As  the  absolute 
value  of  a  positive  number  increases,  its  actual  value  also 
increases,  but  as  the  absolute  value  of  a  negative  number 
i7icreaseSj  its  actual  value  decreases. 

28.  Algebraic  Addition  and  Subtraction  of  Integers.— 

-f-  4  or  simply  4  means  the  number  obtained  by  beginning 
at  zero  and  counting  four  steps  forward,  and  —  4  means 
the  number  obtained  by  beginning  at  zero  and  counting 
four  steps  backward. 

In  general  -\-  a  or  a  means  the  number  obtained  by  be- 
ginning at  zero  and  counting  a  steps  forward,  and  —  a 
means  the  number  obtained  by  beginning  at  zero  and 
counting  a  steps  backward. 

6  -|-  (-[-  4)  m.eans  the  operation  of  beginning  at  plus  6 
on  the  scale  and  counting  four  steps  forward,  or  in  the 
direction  indicated  by  the  sign  of  the  number  to  be  added. 

6  +  ( —  4)  means  the  operation  of  beginning  at  plus  6 
on  the  scale  and  counting  four  steps  backward. 


24  NEGATIVE  QUANTITIES. 

6  —  (+4)  means  the  operation  of  beginning  at  plus  6 
on  the  scale  and  counting  four  steps  backward,  or  in  the 
opposite  direction  to  that  indicated  by  the  sign  of  the 
number  to  be  subtracted. 

6  —  (—  4)  means  the  operation  of  beginning  at  plus  6 
on  the  scale  and  counting  four  steps  forward,  or  in  the  op- 
posite direction  to  that  indicated  by  the  sign  of  the  number 
to  be  subtracted. 

Note.  6  +  (+  4)  and  6  +  (—  4)  having  the  meanings 
given,  which  are  really  definitions  of  addition  of  a  positive 
and  a  negative  quantity,  6  —  (+  4)  and  6  —  (—  4)  must 
have  the  meanings  given  them  because  of  subtraction  being 
the  inverse,  or  opposite,  of  addition. 

In  general,  the  placing  of  one  number  after  another 
with  a  plus  sign  between  indicates  the  operation  of  begin- 
ning on  the  scale  at  the  first  of  the  two  numbers  and 
counting  as  many  steps  as  there  are  units  in  the  number  to 
be  added  and  in  the  direction  indicated  by  the  sign  of  that 
number. 

The  placing  of  one  number  after  another  with  a  minus 
sign  between  indicates  the  operation  of  beginning  on  the 
scale  at  the  first  of  the  two  numbers  and  counting  as  many 
steps  as  there  are  units  in  the  number  to  be  subtracted,  and 
in  the  opposite  direction  to  that  indicated  by  the  sign  of 
that  number. 

EXERCISE  X. 

Find  by  actual  counting  on  the  scale  the  values  of  the 
following  expressions : 

I. 

1.     12 +  (+6):  2.  12 +  (^6). 

3.       6 +  (+12).  4.  -6  +  (+12> 

5^       6 +  (--12).  6,  -12 +  (+6), 


NEGATIVE  QUANTITIES.  25 


7. 

-  6  +  (-  12). 

8. 

-12 +  (-6). 

9. 

12  -  (-  6). 

10. 

-  12  -  (-  6). 

11. 

4  -  (+  4). 

12. 

4 +(-4). 

13. 

«  -  (+  a). 

14. 

«+(-«). 

15. 

-  6  -  (+  12). 

16. 

-  6  -  (-  12). 

17. 

a  ~  (-  a). 

18. 

-  ft  -  (+  ft). 

19,     Designate  the  pairs  of  operations  above  which  give 
precisely  the  same  result. 

29.  Corresponding  Positive  and  Negative  Numbers. — 

Every  positive  number  in  algebra  has  a  corresponding  neg- 
ative number,  that  is,  a  number  the  same  distance  from 
zero  on  the  opposite  side. 

The  sum  of  a  positive  number  and  its  corresponding 
negative  number  is  zero.     Thus, 

6  +  (-  6)  =  0,     ft  4-  (-ft)  =  0. 

30.  Special  Signs  of  duality. — To  indicate  whether 
the  number  to  be  added  or  subtracted  is  positive  or  nega- 
tive, instead  of  enclosing  the  number  with  an  ordinary  plus 
or  minus  sign  before  it  within  a  parenthesis,  we  may  simply 
put  a'  small  plus  or  minus  sign  before  the  number  at  the 
top,  and  when  the  number  is  positive  the  small  plus  sign 
may  be  omitted.     Thus, 

ft  4-  (+  ^)  may  be  written  a  -{-  '^h  or  a-\-  i. 

a-\-  (—  b)  may  be  written  a  +  ~b. 

ft  —  (+  ^)  may  be  written  ft  —  '•"J  or  ft  —  d. 

a  —  {—  b)  may  be  written  a  —  ~h, 

—  «—(—&)  may  be  written  ~a  —  "b, 
etc. 


26  NEGATIVE  QUANTITIES. 

To  indicate  that  the  a  and  h  may  represent  either  posi- 
tive or  negative  numbers  we  may  write  ^a  -\-  "^b. 

31.  Commutative  Law  of  Addition. — From  examples 
1  and  3  in  Exercise  X  we  see  that  a  -\-  b  =  b  -\-  a;  from 
examples  7  and  8,  that  ~a  -\-  ~b  =  'b  -\-  ~a;  from  examples  5 
and  6,  that  ~a  -\-  b  =  b  -{-~a;  and  from  examples  2  and  4, 
that  a  -\-  ~b  =   ~b  -\-  a. 

Whence  we  have  the  following  general  law : 

^a  +  ''b=  ^Z*  +  ="«. 

In  words,  the  algebraic  sum  of  two  numbers  is  the  same 
no  matter  in  what  order  the  numbers  are  taken. 

This  is  known  as  the  Commutative  Law  of  Addition. 

32.  Addition  and  Subtraction  of  Corresponding  Num- 
bers.— Show  by  actual  counting  on  the  algebraic  scale  that 

8  +  -4  =:  8  -  +4,     or     8-4  =  4 

-8  +  +4  =   -8  -   -4  =   -4. 
Also  that 

8  +  +4  =  8-   -4  =  12 
and  -8  +  +4  =   -8  -  -4  =  -  4. 

In  general, 

^a  +  -b  =  ^a  -  n,     or     *«  -  5 
and  "-a  +  +Z*  =  *r/  -  "6. 

Whence     ="«  -f  H^  *a  -  n. 

In  words,  tlie  addition  of  any  number  has  precisely  the 
same  effect  as  the  subtraction  of  the  corresponding  number 


NEGATIVE  QUANTITIES.  27 

toith  the  reverse  sign.  And  the  subtraction  of  any  number 
has  precisely  the  same  effect  as  the  addition  of  the  corre- 
sponding number  with  the  reverse  sign.  This  is  one  of  the 
most  important  theorems  of  algebra. 

33.  Associative  Law  of  Addition.  —  Show  by  actual 
counting  on  the  algebraic  scale  that 

8  +  5-3-4  =  (8 +  5) -3- 4, 

•        =  8  +  (5  -  3)  -  4, 

=  (8  -f  5  -  3)  -  4, 

=  8  + (5 -3 -4), 

===  8  +  5  -  (3  +  4)  =  6. 

In  general, 

-^^  +  ^b-  ^c-  *^  =  (-=«  +  n)  -  ^c-  ^d, 
=  *«  4-  (^b  -  ^c)  -  "-d, 
=  {^a-\-  ^b-  ^c)  -  ^d, 
=  =^«  +  (*^  -  ^c  -  ^-d), 

=  ^a-]-  H-  (^6?+  ^d). 

In  words,  the  sum  of  three  or  more  numbers  is  the  same 
in  whatever  way  the  numbers  may  be  aggregated.  This  is 
known  as  the  Associative  Laiv  of  Addition. 

N.B. — When  terms  are  associated  with  a  negative  sign 
before  the  sign  of  aggregation,  the  signs  of  all  the  terms 
within  the  sign  of  aggregation  must  be  reversed.     (21.) 

34.  Oppositeness  of  Positive  and  Negative  Numbers. — 

Positive  and  negative  signs  always  imply  oppositeness.  In 
case  of  abstract  numbers,  a  negative  number  is  simply  the 
opposite  of  a  positive  number;  that  is,  a  number  which 


L 


28  NEGATIVE  QUANTITIES. 

would  produce  zero  when  added  to  its  corresponding  posi- 
tive number.  Positive  and  negative  numbers  always  tend 
to  cancel  each  other. 

In  the  case  of  concrete  numbers,  a  negative  number  is 
the  result  of  a  measurement  in  the  opposite  direction  to 
that  which  gives  a  positive  number. 

Thus,  distances  measured  to  the  right  or  upward  are 
usually  regarded  as  positive,  and  those  measured  to  the  left 
or  downward  as  negative.  Dates  after  a  certain  era  are 
regarded  as  positive,  and  those  before  the  era  as  negative. 
Degrees  of  temperature  above  zero  are  positive,  while  those 
below  zero  are  negative. 

Assets  are  usually  regarded  as  positive,  and  debts  as 
negative. 

A  surplus  is  positive,  and  a  deficiency  negative. 

The  following  quotation  is  from  Dupuis'  Principles  of 
Elementary  A Igehra : 

"It  an  idea  which  can  be  denoted  by  a  quantitative 
symbol  has  an  opposite  so  related  to  it  that  one  of  these 
ideas  tends  to  destroy  the  other  or  to  ronder  its  effects  nu- 
gatory, these  two  ideas  can  be  algebraically  and  properly  . 
represented  only  by  the  opposite  signs  of  algebra. 

'^  If  a  man  buys  an  article  for  b  dollars  and  sells  it  for  s 
dollars,  his  gain  is  expressed  by  s  —  b  dollars.  So  long  as 
s  >  b,  this  expression  is  -f,  and  gives  the  man's  gain. 

^'But  if  s  <  b,  the  expression  is  — .  It  denotes  that 
whatever  his  gain  is  now,  it  is  something  exactly  opposite 
in  character  to  what  it  was  before.  And  as  he  now  sells 
for  less  than  he  buys  for,  he  loses.  In  other  words,  a  neg- 
ative gain  means  loss. 

"  Thus,  gain  and  loss  are  ideas  which  have  that  kind 
of  oppositeness  which  is  expressed  by  oppositeness  in  sign. 
If  a  man  gains  +  a  dollars,  he  is  so  much  the  wealthier:  if 
he  gains  —  a  dollars,  he  is  so  much  the  poorer. 

**  Whether  gain  or  loss  is  to  be  considered  positive  must 


NEGATIVE  qUANTITIES.  29 

be  a  matter  of  convenience,  but  only  opposite  signs  can 
denote  the  opposite  ideas. 

' '  Among  the  ideas  which  possess  this  oppositeness  of 
character  are  the  following : 

'^  (1)  To  receive  and  to  give  out;  and  hence,  to  buy  and 
to  sell,  to  gain  and  to  lose,  to  save  and  to  spend,  etc. 

'•  (2)  To  move  in  any  direction  and  in  the  opposite  direc- 
tion; and  hence,  measures  or  distances  in  any  direction 
and  in  the  opposite  direction,  as  east  and  west,  north  and 
south,  up  and  down,  above  and  below,  before  and  behind, 
etc. 

^^(3)  Ideas  involving  time  past  and  time  to  come;  as, 
the  past  and  the  future,  to  be  older  and  to  be  younger  than, 
since  and  before,  etc. 

"(4)  To  exceed  and  to  fall  short  off;  as,  to  be  greater 
than  and  to  be  less  than,  etc." 

EXERCISE  XI. 

Give  the  meaning  of  the  following  expressions: 


1.  —  6  A.D.  2.     ~n  A.D. 

3.  "40  B.C.  4.      ~«  B.C. 

5.  —  (—  30)  B.C.  6.      —  ~h  B.C 

7.  —    "50  A.D.  8.       —  (—  c)  A.D, 

9.  The  temperature  is  —  20°. 

10.  The  temperature  has  risen  —  12°. 

11.  The  temperature  has  fallen  —  16°, 

12.  The  temperature  has  fallen  —  (  —  7°). 

13.  The  temperature  has  fallen  —  ~8°„ 

14.  The  temperature  has  risen  ~  ~«°o 


30  NEGATIVE  QUANTITIES. 

15.  It  is  —  17°  colder  to-day  than  yesterday. 

16.  It  is  —  8°  warmer  to-day  than  yesterday. 

17.  It  is  —  ~12°  warmer  to-day  than  yesterday. 

18.  Howard  lives  —  3  miles  east  of  Albert. 

II. 

19.  Louis  lives  —  5  miles  north  of  Horace. 

20.  Ethel  is  —  4  years  older  than  Edith. 

21.  Mabel  is  —  6  years  younger  than  Florence. 

22.  Hilda  is  —  (—  2)  years  younger  than  Margaret. 

23.  Hermon  owes  the  grocer  —  3  dollars. 

24.  Hilda  weighs  —  7  pounds-more  than  Louis. 

26.  Mr.  Crane  is  —  20,000  dollars  richer  than  Mr^ 
Weston. 

EXERCISE  Xil. 

1.  A  man  having  c  dollars  paid  out  a  dollars  to  one 
person  and  h  dollars  to  another.  Express  in  two  ways  what 
he  had  left. 

2.  A  man  bought  at  a  market  tomatoes  at  a  cents  a 
peck  and  potatoes  at  h  cents  a  peck,  and  paid  7n  cents  for 
an  equal  number  of  pecks  of  each.     How  many  pecks  did 

he  buy  ? 

3.  Two  cities  are  42  miles  apart.  Two  men  start  at 
the  same  time  from  the  two  cities  and  walk  towards  each 
other.  The  first  travels  four  miles  an  hour  and  the  second 
three  miles  an  hour.  In  how  many  hours  will  they  meet 
and  how  far  will  each  have  travelled  ? 

4.  Two  cities  are  a  miles  apart.  Two  men  start  at  the 
same  time  from  the  two  cities  and  travel  towards  each 


NEGATIVE  QUANTITIES.  31 

other,  the  first  at  the  rate  of  m  miles  an  hour,  and  the 
second  at  the  rate  of  n  miles  an  hour.  In  how  many  hours 
will  they  meet,  and  how  far  will  each  have  travelled  ? 

6.  Find  two  numbers  whose  sum  is  108  and  such  that 
10  times  the  greater  minus  5  times  the  less  shall  be  less 
than  762  by  4  times  the  sum  of  the  numbers. 


CHAPTER  IV. 

ADDITION    OF    INTEGRAL    ALGEBRAIC 
EXPRESSIONS. 

35.  Arithmetical  and  Algebraic  Sums. — The  sum,  or 
amount,  of  two  or  more  integral  numbers  is  the  number 
obtained  by  counting  all  the  numbers  together.  The  oper- 
ation of  finding  the  sum  of  two  or  more  numbers  is  called 
aclditio7i. 

Since  the  numbers  of  arithmetic  are  all  positive,  the 
addition  of  a  number  in  arithmetic  will  always  increase  the 
number  of  units  in  the  number  to  which  the  addition  is 
made,  and  the  sum  of  two  or  more  numbers  will  contain  as 
many  units  as  all  the  numbers  together.  The  arith7netic 
sum  of  two  or  more  numbers  is  the  sum  of  the  numbers 
without  regard  to  their  signs.  That  is,  it  is  the  sum  of  the 
absolute  values  of  the  numbers. 

In  algebra,  the  addition  of  a  positive  and  a  negative 
number  will  tend  to  diminish  the  number  of  units  in  the 
number  which  has  the  greater  absolute  value.  The  alge- 
braic sum  of  two  such  numbers  is  the  arithmetical  difference 
of  the  numbers  with  the  sign  of  the  one  which  has  the 
larger  absolute  value. 

The  algebraic  sum  of  two  numbers  both  positive  or  both 
negative  is  the  arithmetic  sum  of  the  numbers  with  then 
common  sign.     Thus, 

8  +  10  =  18,       -8  +  -10  =  -  18, 

8  +  -10  rr:   ~  2,  "8  +  10  =  +  2. 

32 


ADDITION  OF  INTEGERS.  33 

The  algebraic  sum  of  two  or  more  numbers  is  the  sum 
of  the  numbers  regard  being  had  to  their  signs.  That  is, 
it  is  the  sum  of  the  actual  values  of  the  numbers. 

36.  Signs  of  Coefficients. — The  sign  of  a  term  may  be 
regarded  as  belonging  to  its  coefficient  only.  That  is,  plus 
terms  may  be  regarded  as  those  whose  coefficients  are  posi- 
tive. The  reason  for  this  will  appear  farther  on,  under 
Multiplication. 

37.  Integral  Algebraic  Expressions.  —  It  has  been 
learned  in  arithmetic  that  numbers  are  not  only  integral, 
but  also  fractional  and  surd.  In  any  algebraic  expression 
the  letters  may  stand  for  any  kind  of  number. 

An  algebraic  expression  such  as 

x^  +  6x^  -  4:x^  -  3x^  +  2:c  +  1, 
or         .     ,      l-{-2x-3x^  -  4:X^  +  5x^  +  x^, 

in  which  the  exponents  of  the  letters  are  all  positive  inte- 
gers, and  in  which  none  of  the  letters  occur  in  the  denom- 
inators of  fractions,  or  in  the  divisors  of  an  indicated 
division,  are  called  ifitegral  algebraic  expressions.  The  co- 
efficients of  the  various  terms  may  be  fractional. 

38.  Extension  of  the  Application  of  the  Formal  Laws 
of  Addition. — In  the  addition  of  integral  algebraic  expres- 
sions it  is  assumed  that  the  commutative  and  associative 
laws  already  established  for  integral  numbers  apply  equally 
to  fractional  and  surd  numbers.  This  is  in  accordance 
with  the  generalizing  spirit  of  algebra. 

39.  Definition  of  Addition  of  Algebraic  Expressions. — 

To  add  integral  algebraic  expressions  is  to  combine  their 
various  terms  into  a- single  algebraic  expression,  each  term 
to  be  preceded  by  its  own  proper  sign.  The  resulting  ex- 
pression should  be  given  in  its  simplest  form. 


34  ADDITION  OF  INTEGERS. 

40.  Addition  of  Monomials  and  Polynomials. — Similar 
terms  are  analogous  to  concrete  numbers  of  like  denomina- 
tions, and  dissimilar  terms  are  analogous  to  concrete  num- 
bers of  unlike  denominations. 

Similar  terms  may  be  added  by  finding  the  algebraic 
sum  of  their  coefficients  and  writing  after  this  the  common 
literal  factors  of  the  terms.  Thus,  the  sum  of  ba^b,  1la%, 
and  —  ^a^h  is  4a^^. 

Dissimilar  terms  can  be  added  only  by  placing  them 
one  after  another  in  a  polynomial  expression  each  with  its 
own  sign.  Thus,  the  sum  of  ^a%,  —  4:al),  and  5c  is  3a^  — 
4:ab  +  5c.  The  sum  of  these  dissimilar  terms  is  really 
3a^  +  ~^(ib  -\-  5c,  but,  as  we  have  seen,  to  add  ~4:ab  is  the 
same  as  to  subtract  +4<x&,  or  +  ~'4:ab  =  —  4:ab. 

The  following  examples  will  illustrate  the  working 
rules  of  addition : 

Ex.   1.       3A  -7b^y 

7a^x  —  9b^y 

5  A  —  5b^y 


15a^x  -  21Py 

To  add  similar  terms  with  like  signs,  an?iex  the  common 
literal  factors  to  the  arithmetical  sum  of  the  coefjicients,  and 
prefix  the  common  sign. 

Ex.   2. 


7^y 

9>abx 

^xY 

—  \ahx 

~  QxY 

-  %abx 

Wy^ 

IQabx 

^  9^y 

■—  7abx 

—    5(?^y^  l%abx 


ADDITION  OF  INTEGERS.  35 

To  add  similar  terms  tcith  unlike  signs,  find  the  arith- 
metical sum  of  the  coefficients  of  the  plus  terms,  and  of  the 
coefficients  of  the  minus  terms,  and  tlie  arithmetical  differ- 
ence of  these  tivo  sums,  anyiex  to  this  difference  the  common 
literal  factors,  and  prefix  the  common  sign  of  the  terms 
whose  coefficients  produce  the  larger  arithmetical  sum. 

Ex.    3.       a  Sax 

b  —  4tby 

—  c  —hd 


a  -\-  b  —  c  Sax  —  4:bg  —  5d 

To  add  dissimilar  terms,  write  them  one  after  another, 
each  ivith  its  oivn  sign. 

Ex. 


4.       —a 

Sx'y 

-b 

-7x^g 

da 

-Qxy^ 

-2b 

-  Sb 

-5 

-'Sxy^ 

2a-db-  5 

-    4:Xh/    -    QXlf    - 

■  Sb 

To  add  terms  some  of  which  cere  similar  a7id  some  dis- 
similar, combine  the  different  sets  of  similar  terms  into 
single  terrns,  and  write  the  resulting  terms  together  icith 
the  remaining  terms  one  after  another  in  a  2^oly7io7nial  ex- 
pression each  icith  its  own  sign. 

Ex.    5. 


2cd-    3cx^  +  2c^x 

-  Scd  -      ex'  -  5c^x  -f-  cx^ 

12cd  +  lOcx'  -  Qc'x           - 

11 

Qcd  +    Qcx'  -  9c'x  +  cx^  -  11 
To  add  polynomials,  combine  the  different  sets  of  similar 


36  ADDITION  OF  INTEGERS. 

terms  in  the  polynomials  into  si^igle  terms,  and  write  these 
and  the  remaining  terms  as  a  polynomial. 

In  the  addition  of  polynomials,  it  is  convenient  to  ar- 
range the  terms  so  that  the  similar  terms  will  fall  in  verti- 
cal columns. 

41.  Simplification  of  Polynomials. — When  any  polyno- 
mial contains  one  or  more  sets  of  similar  terms,  it  may  be 
simplified  by  combining  these  sets  into  single  terms. 

EXERCISE  XIII. 

Find  the  sum  of  the  following  terms: 

I. 

1.  3«,  7«,  2«,  a,  12a. 

2.  7a^x,  da^x,  c^x,  20a^x. 

8.  -  6ab^  -  ab\  -  lal)\  -  llaJ)%  -  4.ay^,  -  8«5l 

4.  —  Ix,  —  2x,  —  8x,  —  X,  —  12Xf  —  llic,  —  15a;. 

6.  Zx^,  -  bx\  82^2,  -  12x^. 

6.  —  bac^x,  ac^x,  —  %a(?x,  \^a(?x. 

7.  5?/2,  4«c,  —  ac,  —  7y^,  —  6ac,  iy"^,  —  5. 

8.  7a^x.  —  Aad,  —  ax^,  —  da^x,  —  8,  —  5abo 
Simplify  the  following  polynomials : 

I. 

9.  4:X  —  5al)  -]-  7x  -{-  c  -\-  llab  —  20a;. 

10.  daH^  -7x^-  5-{-  12x^  -  4:a^^  +  12  -  c. 

11.  l/'dx  -  l/2x  +  d/4cx  +  X. 

12.  2/'6y  -  S/4.y  -  2y  -  l/3y  +  6/6y  +  y. 

13.  9(a  +  5)  +  10{a  +  ^)  -  («  +  ^)  ~  2{a  +  5). 


ADDITION  OF  mTEGBm.  Zl 

II. 

14.  7«  -  3(^  +  2^)  +  8«  -  (a;  +  2/)  +  3(a:  +  y)  -  16«. 

15.  2(m  +  n)  +  3(a  +  Z»)  +  («  +  ^»)  -  {m  -f  /^)  + 
(r^  +  /;)  -  6(m  +  7^. 

16.  3r/,(/;  +  a:)  +  5r/(^'  +  a;)  +  r^C^*  ^  x)  -  lla{b  +  a;). 

17.  2C(«2  _  J2)  _  3^(^2  _  J2)  _^  6^.(^2  _  ^2)  _  ic(^2  _  J2). 

Add  the  following  polynomials: 

I. 

18.  Mz  —  ^hy  -  8,  -  2az  +  bhy  +  6,  6az  +  6  J?/  -  7, 
and  —  Mz  —  7%  +  ^^ 

19.  Soa:;  —  dcz^,    —  5ax  +  5c;2;^   ax  +  2c2;^,   and  —  iax 

—    4:CZ^. 

20.  8^,  +  h,2a-i  +  c,-  3«  +  SZ*  +  2^,  -  65  -  3c 
+  dd,  and  —  5«  +  7c  —  2d 

II. 

21.  7a:  —  G?/  +  5;^  +  3  —  ^,  —  a;  —  3?/  —  8  —  ^,  —  a: 
+  2/  ~  '"^^  -  1  +  '^ff^    -  2x  ^  dy  -{-  3z  -  1  -  g,    and  a;  + 

Sy-5z  +  ^+g. 

22.  2«'^  +  5ffZ>  -  xy,  -  7a^  +  dab  -  dxy,  -  3a^  - 
^ab  -\-  6xy,    and   9a^  —  ab  —  %xy, 

23.  ^a^b^  -  MV^  +  x}y  +  xa/.  ia^'^  -  la^  -  dxif  + 
6x2?/,     3^^3^2  _^  3^^2^3  _  3^2^  _|_  5,^^2^     ^nd     '^a^b^  -  a%^  - 

dx'^y  —  '6xy'\ 

I. 

24.  A  lady  bought  three  yards  of  ribbon  at  a  cents  a 
yard,  10  yards  of  tape  at  c  cents  a  yard,  and  five  spools  of 
thread  at  d  cents  a  spool.  She  paid  x  cents  on  the  bill. 
How  much  remains  due  ? 


38  ADDITION. 

25.  One  morning  tlie  mercury  in  the  thermometer 
stood  at  X  degrees.  During  the  next  24  hours  it  rose  h  de- 
grees and  fell  c  degrees.  The  following  day  it  rose  d  de- 
grees.    What  was  its  height  then  ? 

26.  A  father  divided  his  property  of  27,000  dollars 
among  his  four  children,  giving  500  dollars  less  to  each  in 
succession  from  the  eldest  to  the  youngest.  How  much  did 
he  give  to  each  ^ 

ir. 

27.  A  father  gave  his  eldest  son  x  dollars,  his  second 
son  7  dollars  less,  his  third  son  9  dollars  less  than  the  sec- 
ond, and  his  fourth  son  1 1  dollars  less  than  the  third.  How 
much  did  he  give  to  all  ? 

28.  A  father  divided  his  property  among  his  four  chil- 
dren. To  each  of  the  first  three  he  gave  1/4  of  his  prop- 
erty plus  200  dollars,  and  to  the  fourth  he  gave  1400  'dol- 
lars.    What  was  the  value  of  his  property  ? 

29.  A  man  left  his  five  children  x  bonds  worth  a  dol- 
lars each,  and  x  acres  of  land  worth  i  dollars  each ;  but  he 
owed  m  dollars  to  each  of  q  creditors.  W^hat  was  each 
child's  share  of  the  estate  ? 

42.  Aggregation  of  Coefficients. — When  two  or  more 
terms  of  a  polynomial  contain  one  or  more  common  factors, 
whether  numeral  or  literal,  the  terms  may  be  collected  into 
one  by  enclosing  the  terms  within  a  parenthesis  and  placing 
the  common  factors  outside. 

When  the  common  factors  are  numeral  and  literal,  it  is 
customary  to  place  the  numeral  factor  and  the  letters  which 
belong  to  the  first  part  of  the  alphabet  before  the  parenthe- 
sis, and  the  letters  which  belong  to  tlie  last  part  of  the 
alphabet  after  the  parenthesis. 

e.g.        bacx  +  bbcx  —  bcdx  =  bc{a  -{-  h  —  d)x. 


ADDITION  OF  INTEOEBS  39 

EXERCISE  XIV. 

Collect  the  coefficients  of  x  and  y  in  tho  following  ex- 
pressions : 

I. 

1.  ax  -\-hy  -\-  mx  -|-  ny. 

2.  7nnx  -\-  2by  -{-  pqx  —  4:by. 

3.  3x  —  2y  -{-  (Jbx  —  4//  +  7ax  -\-  m  -\-  n. 

4.  ^ax  +  ^hx  +  hy  +  1x  -  by  ^  x  -  5y. 

5.  Howard  is  twice  as  old  as  Albert.  If  x  represents 
Albert's  age  now,  what  would  represent  their  respective 
ages  eight  years  hence  ? 

6.  Howard  is  now  twice  as  old  as  Albert,  but  12  years 
from  now  he  will  be  only  3/2  as  old.     How  old  is  each  ? 

7.  Two  cities,  A  and  B,  are  on  a  straight  road  and  18 
miles  apart.  Two  couriers,  P  and  Q,  start  at  the  same 
time  from  the  respective  cities  and  travel  in  the  same  direc- 
tion, P  from  A  towards  P  at  the  rate  of  eight  miles  an 
hour,  and  Q  from  B  at  the  rate  of  six  miles  an  hour.  In 
how  many  hours  will  P  overtake  Q,  and  how  far  will  each 
have  travelled  ? 

8.  Divide  the  number  a  into  two  parts,  one  of  which 
shall  exceed  the  other  by  b. 

II. 

9.  ax  -j-  by  -\-  rz  —  mx  —  ny  —  2^^- 

10.  "idx  +  ^ey  +  4A  -  2/:r  -  ?>dy  +  Uz. 

11.  ^/oay  -  2x  +  ^/\by  +  Qax. 

12.  ^ax  —  by  —  3bx  —  4:ay. 


40  ADDITION. 

13.  Horace  is  now  twice  as  old  as  Herbert,  but  a  years 
from  now  he  will  be  only  4/3  as  old.     How  old  is  each  ? 

14.  Two  towns,  A  and  B,  are  a  miles  apart.  Two  cour- 
iers, P  and  Q,  set  out  at  the  same  time  from  the  respective 
towns,  and  travel  in  the  same  direction.  P  travels  from  A 
towards  B  at  the  rate  of  h  miles  an  hour,  and  Q  from  B  at 
the  rate  of  c  miles  an  hour.  In  how  many  hours  will  P 
overtake  Q,  and  how  far  will  each  have  travelled  ? 


CHAPTER  Y. 

SUBTRACTION  OP  INTEGRAL  ALGEBRAIC 
EXPRESSIONS. 

43.  Definition  of  Subtraction. — Subtraction  is  the  in- 
verse of  addition,  or  the  process  of  undoing  the  operation 
of  addition.  In  addition,  two  numbers  are  given  and 
their  sum  or  amount  required.  In  subtraction,  the  sum 
of  two  numbers  and  one  of  the  numbers  are  given,  and 
the  other  is  required. 

The  given  sum  is  called  the  mimie^id,  the  given  num- 
ber the  subtrahend,  and  the  required  number  the  difference 
or  remainder. 

Since  the  minuend  is  the  sum  of  the  subtrahend  and 
difference,  we  may  prove  our  subtraction  by  adding  the 
subtrahend  and  difference  to  see  if  their  sum  agrees  with 
the  minuend. 

44.  Rule  for  Subtraction  of  Integral  Algebraic  Ex- 
pressions.— We  have  already  seen  in  section  15  that  the 
addition  of  any  number  produces  the  same  effect  as  the 
subtraction  of  the  corresponding  number  with  the  reverse 
sign,  or,  conversely,  the  subtraction  of  any  number  is 
equivalent  to  the  addition  of  the  corresponding  number 
with  the  reverse  sign.  Hence  we  have  the  following  rule 
for  algebraic  subtraction : 

Add  the  subtraliend  with  its  signs  reversed  to  the  minu- 
end, 

41 


42  SUBTRACTION. 

In  the  operation  of  subtraction  it  is  better  not  actually 
to  change  the  old  signs,  but  merely  to  think  of  them  as 
changed  in  the  addition.  If  the  new  signs  are  written,  it 
is  better  not  to  change  the  old  into  the  new,  but  to  write 
the  new  as  small  signs  before  the  terms  at  the  top. 

EXERCISE  XV. 
I. 

1.  From  "Zx -\-  y  -\- 1  z  take  6x  -]- 2y  —  7z. 

2.  From  9a  —  4:b  -\-  Sc  take  5a  —  db  -{-  c. 

3.  Subtract  3a^  -  a^^la-U  from  lla^  -  2a^  +  da^ 

—  Sa. 

4.  From  10«V  +  Uax^  +  8A  take  -  lOa^x^  +  15ax^ 

-  8  A. 

5.  Subtract  1  —  a  -\-  a^  —  da^  from  a^  —  1  -\-  a^  —  a. 

6.  From  2/3^:2  _  ^/^x  -  1  take  -  2/3^2  -{- x  -  1/2. 

7.  From  a  take  b  —  c. 

8.  What  must  be  taken  from  6a  +  5  —  3J  to  produce 

8a  +  6Z»  +  13  ? 

9.  What  must  be  taken  from  2x^  —  3a^x^  +  9  to  pro- 
duce x^  -\-  ba^x^  —  3  ? 

10.  What  must  be  added  to  a  +  5Z»  +  9  to  produce 
3a  -  2^*  +  6  ? 

11.  Ethel  is  twice  as  old  as  Edith,  and  six  years  ago 
she  was  four  times  as  old.     What  is  the  age  of  each  ? 

12.  A  and  B  have  together  150  dollars.  If  A  were  to 
give  ^35  dollars,  B  would  have  three  times  as  much  as  ^. 
How  much  has  each  ? 

II. 

13.  What  must  be  added  to  x  to  produce  y  ? 


PARENTHESES.  43 

14.  By  how  much  does  6x  —  7  exceed  3a;  +  4  ? 

15.  From  what  must  5a;  -f-  4?/  +  7ft  —  12  be  subtracted 
to  produce  unity  ? 

16.  From  what  must  x^  —  x^  -\-  x  —  IhQ  subtracted  to 
produce  %x'^  -|-  2  ? 

17.  From  l{a  +  I)  take  3(ft  +  h), 

18.  From  3«(6'  —  x)  take  a{c  —  x). 

19.  From  la^(l)  —-x)  —  ab(a  —  b)  take  5a^{h  —  x)  — 
bab(a  —  h). 

20.  Howard  is  x  years  old.  How  old  was  he  eight  years 
ago? 

21.  Divide  the  number  m  into  two  parts  such  that, 
when  a  is  taken  from  the  first  and  given  to  the  second,  the 
second  will  be  five  times  the  first. 

PAREKTHESES. 

45.  Operation  upon  Aggregates. — Every  algebraic  ex- 
pression, however  complex,  represents  a  quantity,  and 
may  be  operated  upon  as  if  it  were  a  single  symbol  of  that 
quantity. 

When  an  expression  is  to  be  operated  upon  as  a  single 
quantity  it  is  enclosed  within  parentheses  or  brackets,  but 
the  parenthesis  may  be  omitted  when  no  ambiguity  or  error 
will  result  from  the  omission. 

Thus,  one  polynomial  may  be  added  to  another  or  to  a 
monomial  by  writing  it,  enclosed  within  a  parenthesis  and 
preceded  by  a  plus  sign,  after  the  expression  to  which  it  is 
to  be  added;  and  a  polynomial  may  be  subtracted  from  a 
polynomial  or  monomial  expression  by  writing  it,  enclosed 
within  a  parenthesis  and  preceded  by  a  minus  sign,  after 
the  expression  from  which  it  is  to  be  subtracted. 

Since  terms  written  after  one  another  each  with  its  own 


44  SUBTH ACTION. 

sign  in  a  polynomial  expression  are  to  be  considered  as 
added,  and  since  in  addition  there  is  no  change  of  signs,  a 
parenthesis  preceded  by  a  plus  sign  may  be  omitted  without 
any  change  of  signs;  and  since  the  subtraction  of  any 
quantity  produces  the  same  effect  as  the  addition  of  the 
corresponding  quantity  with  the  reverse  sign,  a  parenthesis 
preceded  by  a  minus  sign  may  be  omitted  if  the  sign  of 
every  term  be  changed. 

N.B". — It  must  be  carefully  borne  in  mind  that  the  sign 
before  the  parenthesis  is  not  the  sign  of  the  first  term 
within  it,  but  of  the  parenthesis  as  a  whole.  This  sign 
really  goes  with  the  parenthesis  when  the  latter  is  removed. 
When  no  sign  is  expressed  with  the  first  term  within  the 
parenthesis,  the  term  is  understood  to  be  plus,  and  its  sign 
must  be  written  on  the  removal  of  the  parenthesis,  as  plus 
when  the  parenthesis  is  plus,  and  as  minus  when  the 
parenthesis  is  minus. 

EXERCISE    XVI. 

Clear  the  following  expressions  of  parentheses  and  re- 
duce the  results  to  the  simplest  form : 


1. 

I. 

ah  —  (m,  —  'Sal)  +  2ax)  —  7ab. 

2. 

X  —  {a  —  x)  -\-  (x  —  a). 

3. 

2b+{h-  2c)  -  {b-\-  2c). 

4. 

4:X  -  3y  -\-  2z  -  {-  7x  +  5y  -  Sz 

5. 

II. 

7ax  —  2%  —  {8ax  +  Sbi/)  —  {Sax  ■ 

6. 

{a  —  x)  —  (a  -{-  x)  -\-  2x. 

7. 

-(a-b)-(b-r)-(,-  a). 

8. 

-  (Sm  +  2»)  -  {3m  -  2n)  +  9m. 

Zby) 


PARENTHESES.  45 

22.  Of  course  in  forming  aggregates  preceded  by  a 
minus  sign,  the  sign  of  every  term  enclosed  within  the 
parenthesis  must  be  changed. 

EXERCISE  XVII. 

Keduce  the  following  expressions  to  the  form  x  —  (an 
aggregate) : 

I. 


1. 

X  —  a  —  h. 

2. 

X  —  7)1  —  n. 

3. 

a-{-  X  —  'dx-\-^y. 

4. 

-  3^  +  a;  4-  2c  +  56?. 

6. 

2x-2a-i-  2b. 

6. 

x  +  S  -  (a-\-b). 

7. 

X  -{-  a  —  (b  ~  o)  -{-  (m  —  n). 

II. 

8. 

2x  -\-  a  —  b. 

9. 

3x  -  2m  +  2n. 

10. 

ox  -{-  ab  —  m  —  Sab  -\-  2m. 

11. 

X  —  2m  —  {Sa  —  2b). 

12. 

X  —  {am  -\-  b)  —  {p  —  q)  —  (am  —  n). 

13 

X  —  (a^b)  —  (p  —  q)  —  (m  —  n). 

46. 

Compound    Parentheses. — An  algebraic   expression 

having  parentheses  as  a  part  of  it  may  be  itself  enclosed 

in  parentheses  with  other  expressions,  and  this  may  be 
repeated  to  any  extent.  Each  order  of  parentheses  must 
then  be  made  larger  or  thicker,  or  different  in  shape,  to 
distinguish  it. 

e.g.  Suppose  we  have  to  subtract  a  from  b,  the  remain- 


46  SUBTRACTION. 

der  from  c,  that  remainder  from  d,  and  so  on.     We  shall 
have : 

First  remainder, h  —  a. 

Second  remainder, c  —  {h  —  a). 

Third  remainder,  .     o     .     .    d  —  [^c  —  {h  —  a)]. 
Fourth  remainder,      .     e  —  {d  —  \g  —  {i  —  a)]]. 
Fifth  remainder,     ^  —  [e  —  \d  —  [^  —  (^  —  «)] }]. 

Such  parentheses  are  called  compoimd  pare^itJieses. 

Compound  parentheses  of  addition  and  subtraction  may 
be  removed  by  removing  separately  the  individual  paren- 
theses of  which  they  are  composed.  AVe  may  begin  either 
with  fhe  outer  ones  and  go  inward,  or  with  the  inner  ones 
and  go  outward.   It  is  customary  to  begin  with  the  inmost. 

e.g.  Clear  of  parentheses: 

^_[«_  {j_  [c_  (^_e)]}]. 

Beginning  with  the  inmost,  the  expression  takes,  in 
succession,  the  following  forms : 

ic-  [«-  |&-  [c-  ^  +  e]j]  = 
X  —  \a  —  [h  —  c -\-  d  —  e}']  = 
X—  [a  —  I)-{-G  —  d-{-e]  = 
X  —  a  -{■  b  —  G  -\-  d  ~  e. 
Beginning  with  the  outmost,  we  have 

x-[a-  \b-  [G-(d-  e)]}]  = 
x-a-\-{l)-{c-  {d-e)']}  = 
X  —  a~\-h  —  \_c  —  {d  ~  e)']  = 
X  —  a-\-l)  —  G-{-{d—  e)=. 
X  —  a  -\-  b  —  G  -\-  d  —  e. 
Again,         x  —  {—  (a -]- b)  -\-  {c  -\-  d)  —  {e  —  z)"] 


PARENTHESES.  47 

gives,  when  we  begin  with  the  inner  parentheses, 

X  —  \^—  a  —  h  -\-  c  -\-  d  —  e  -\-  z\  = 

x-{-a-\-b  —  c  —  d -\-  G  —  z\ 
and  when  we  begin  with  the  outer  parentheses, 

x^-  {a-^h)  -  {c^  d)  ^  {e  -  z)  =^ 

x-\-a-\-h  —  c  —  d-^-e  —  z, 

EXERCISE  XVIII. 

Remove  the  parentheses  in  the  following  expressions, 
and  combine  the  terms  containing  x,  y,  and  zi 

I. 

1.  rn-{-i-{p-q)-^{a-l)-\-{-c-\-d)]. 

2.  m.-l-  {a-h)-  {p-\-q)-]-  (n  -  h)], 

3.  'Tax  -  [(2ax  +  by)  -  {Sax  -by)  +  (-  "^ax  +  2%)]. 

4.  a—\a—  [a  —\a—  {a  —  «)]}]• 

5.  p—\a  —  h  —  {s-\-t-\-a)-\-{—m  —  n)]. 

6.  A  father  left  80,000  dollars  to  his  four  children.  The 
eldest  was  to  receive  four  times  as  much  as  the  youngest 
less  1800  dollars,  the  second  was  to  receive  three  times  as 
much  as  the  youngest  less  1200  dollars,  and  the  third  was 
to  receive  twice  as  much  as  the  youngest  less  600  dollars. 
How  much  did  each  receive  ? 

7.  Divide  a  into  three  parts  such  that  the  second  shall 
equal  the  first  minus  h  and  the  third  shall  be  c  less  than 
twice  the  first. 

II. 

8.  2aa;  —  \^ax  —  by  —  (7  ax  +  2by)  —  {5ax  —  Sby)]. 

9.  ax  -\-  by  -\-  cz  -\-  [2ax  —  3cz  —  {2cz  -\-  5ax)  —  {7by 
-  dcz)]. 


48  SUBTRACTION. 

10.  X  —  \^x  —  y  —  [3:r  —  %y  —  (4a;  —  3«/)] }. 

11.  ax  —  bz  —  {ax  -\-  bz  —  [ax  —  bz  —  {ax  -f-  bz)]]. 

12.  my  —  \x-}-  Sy  -\-  [2my  —  3{x  —  y)  —  4:ab]  -\-  5]. 

13.  Divide  186  into  five  parts  such  that  the  second 
shall  exceed  the  first  by  12,  the  third  shall  exceed  twice 
the  first  by  24,  the  fourth  shall  exceed  three  times  the  first 
by  36,  and  the  fifth  shall  exceed  four  times  the  first  by  48. 


CHAPTER   VI. 

MULTIPLICATION  OF    INTEGRAL  ALGEBRAIC 
EXPRESSIONS. 

A.  LAW  OF  SIGNS,  OF  COMMUTATION,  AND  OF  ASSOCIATION. 

47.  Multiplication  of  Integers. — Multiplication  is  the 
operation  of  finding  what  number  is  obtained  by  counting 
a  number  over  a  given  number  of  times. 

The  number  to  be  counted  over  is  called  the  multipli- 
cand, the  number  which  indicates  how  many  times  the 
multiplicand  is  to  be  counted  over  is  called  the  multiplier, 
and  the  number  obtained  as  the  result  of  the  operation  is 
called  the  product. 

The  multiplier  and  the  multiplicand  are  called /<a56'^ors 
of  the  product. 

48.  Two  Cases  of  Multiplication.  —  As  there  are  two 
directions  of  counting  from  zero  in  algebra,  so  there  are 
two  cases  of  multiplication.  In  addition,  as  we  have  seen, 
the  numbers  to  be  added  are  counted  in  the  direction  in- 
dicated by  their  signs,  while  in  subtraction  the  numbers  to 
be  subtracted  are  counted  in  the  opposite  direction  to  those 
indicated  by  their  signs.  The  direction  in  which  the  mul- 
tiplicand is  to  be  counted  is  indicated  by  the  sign  of  the 
multiplier.  When  this  sign  is  positive  the  multiplicand  is 
counted  in  the  direction  indicated  by  its  sign.  Hence  the 
sign  in  the  product  will  be  the  same  as  the  sign  in  the 
multiplicand.     When  the  multiplier  is  negative  the  multi- 

49 


50  MULTIPLICATION. 

plicand  is  counted  in  the  opposite  direction  to  that  indicated 
by  its  sign.  Hence  the  sign  is  the  reverse  of  the  sign  in 
the  multiplicand.  The  former  case  corresponds  to  addition 
and  the  latter  to  subtraction.  In  multiplication  the 
counting  is  always  understood  to  begin  at  zero. 

49.  Law  of  Signs  in  Multiplication. — 


Ex.         12  X    4  =  48. 

-12  X     4  =  -48. 

12  X  -4  =  -48. 

-12  X   -4  =:  48. 

4  X  12  =  48. 

4  X-12  =  -48. 

-4  X  12  =  -48. 

-4  X-12  =  48. 

In  general, 

a  xh  =  ah. 

-a  X  h  =  —  ah. 

aX'h  —  —ah. 

-a  x~h  =  ah. 

h  X  a  =  ah. 

h  X~a—  —  ab. 

-h-  X  a  =  —  ah. 

~h  X~a  —  ah. 

From  the  above  we  see : 

1°.  That  like  signs  in  multiplication  produce  jt?/i^s,  and 
unlike  signs  minus. 

2°.  That  interchanging  the  signs  of  the  factors  does  not 
alter  the  sign  of  the  product,     a  X~h  =  —  ah  —~a  Xh. 

3°.  That  interchanging  the  multiplier  and  multiplicand 
does  not  alter  the  product,     a  X~h  —  —  ah  —  ~h  X  a. 

50.  Commutative  Law  of  Multiplication. — From  2°  and 
3°  we  see  that  multipUcation  is  commutative  both  as  re- 
gards its  signs  and  its  factors.  Addition  is  commutative 
only  as  regards  its  terms  and  not  as  regards  its  signs. 

12  +  -4  =  -4  +  12,  but  12  +  -4  does  not  equal  "12  +  4. 

That  multiplication  is  commutative  as  regards  its  fac- 
tors, that  is,  that  the  same  result  will  be  obtained  by  count- 


LAW  OF  ASSOCIATION. 


51 


n 

m 
Fig.  1. 


ing  m  things  over  n  times  as  by  counting  n  things  over  m 
times,  may  be  shown  as  follows. 

Place  711  squares  in  a  horizontal  row  and  repeat  the  row 
vertically  n  times  as  in  Fig.  1. 
Evidently  we  would  get  the  num- 
ber of  squares  in  the  figure  either 
by  counting  the  m  squares  of  the 
"lower  row  over  n  times,  or  by 
counting  the  n  squares  of  the 
left-hand  column  over  m  times. 
Hence  7n  X  n  =  7i  X  m.  Thus 
the  commutative  law  of  multipli- 
cation is  seen  to  be  a  consequence  of  the  associative  and 
commutative  laws  of  addition. 

51.  Associative  Law  of  Multiplication. — In  the  opera- 
tion of  multiplication  we  combine  only  two  factors  at  a 
time  into  a  product.  If  there  are  more  than  two  factors  to 
combine,  we  first  combine  two  of  the  factors  into  a  product, 
and  then  use  the  product  obtained  and  a  third  factor  as  two 
factors  to  form  a  new  product,  and  so  on,  till  the  factors 
are  all  used. 

9  X  3  X  3  =  27  X  2  =  54, 

9X3X2=    9X6  =  54. 


e.g. 


or 


In  general. 


b  .  c  =^  (ab) .  c,    or 


(bo). 


That  is,  the  result  of  multiplying  ahj  b  and  the  prod- 
uct by  c  is  the  same  as  multiplying  a  by  the  product  of  b 
and  c. 

The  fact  that  the  factors  may  be  grouped  or  associated 
in  any  way  is  known  as  the  Associative  Law  of  Multiplica- 
tion. 

The  associative  law  of  multiplication  may  be  shown  to 
be  true  for  integers  as  follows : 


62 


MULTIPLICA  TION. 


m 
Fig.  2. 


Use  the  diagram  of  the  last  section,  and  suppose  each  of 
the  small  squares  to  be  divided 
into  a  rectangles  by  horizontal 
lines  (Fig.  2).  There  will  evi- 
dently be  ma  of  these  small  rect- 
angles in  the  lower  row  of  squares, 
and  na  in  the  left-hand  column, 
and  we  would  get  the  whole  num- 
ber of  rectangles  by  counting  the 
lower  set  of  ma  rectangles  over  n 

times,  or  by  counting  the  lowest  row  of  m  rectangles  over 

na  times.     Hence 

(ma) .  n  =  m .  {na). 

From  the  commutative  law  of  multiplication  we  see 
that  it  makes  no  difference  in  what  order  the  factors  of  a 
product  are  written. 

Hence  the  factors  of  a  term  may  be  written  in  any 
order.  It  is,  however,  customary  to  write  the  numerical 
factor  first  and  the  literal  factors  in  their  alphabetic  order. 

If  there  are  more  than  two  factors,  the  product  will  be 
plus  when  all  the  factors  are  positive,  or  when  the  num- 
ber of  negative  factors  is  even.  The  product  will  be  minus 
when  the  number  of  negative  factors  is  odd. 


e.g.  a  .b  »~c  =  —  ah  .~c  =  ahc, 

~a  .  ~1)  .~c  .  d  ^^  ah  .  ~cd  =  —  ahcd. 


62.  Multiplication  of  Monomials.  —  In  the  multiplica- 
tion of  integral  algebraic  expressions  we  assume  that  the 
laws  of  commutation  and  association  which  we  have  dem- 
onstrated for  integers  also  apply  to  all  numbers  which 
may  be  represented  by  letters,  fractional  and  surd  as  well 
as  integral. 

Hence  we   multiply   two   integral  monomial  algebraic 


LAW  OP  ASSOaiATIOK  53 

expressions  together  by  grouping  all  their  factors  together 
in  a  single  term. 

This  term  must  therefore  contain  every  factor  contained 
in  the  terms  multiplied  together,  and  each  factor  as  many 
times  as  in  all  the  terms  together. 

e.g.     3«2^,sc  X  4:a^b^x  = 

To  multiply  one  monomial  hy  another,  multiply  togefher 
their  numeral  coefficients  and  icrite  after  the  product  ob- 
tained each  letter  of  both  monomials  with  an  exponent  equal 
to  the  sum  of  its  exponents  in  the  two  terms.  Briefly, 
multiply  coefficients  and  add  exponents. 

The  sign  of  the  product  must  be  determined  by  the  law 
of  signs  in  multiplication. 

EXERCISE  XIX. 

Find  the  product  of  the  following  factors: 

I. 

1.  3«   and   7b. 

2.  6a  and   Qa^. 

3.  4:a'^x  and   —  8x^y^. 

4.  a^bx  and   —  a^b^y^. 

6.  —  'iSd'^x^  and   —  H'^x^z^. 

6.  —  Hmhiy^  and    h/nhi^x*. 

7.  am  X  ab  X  ac  X  ad. 

8.  ax  X  ~  bx  X  ex  X  dx. 


54:  MULTIPLICATION, 

9.  X  X  —  ax  X  —  ahx  X  —  abcx, 

10.  ^ax  X  —  '^aH^  X  —  bahnx. 

11.  —  l7n^y  X  —  ^a^y^  X  6ax, 

12.  27n  X  n  X  —  a  X  —  2b. 

13.  —  Sax  X  —  2h7i  X  —  7x  X  —  4:imx, 

14.  —  ny  xgy  X  —  ^  X  3bm. 
16.  xy  X  2y^  X  y^x  X  2ayx^. 

16.  5y^  X  —  3gy  X  —  2x^  x  —  ao^z, 

II. 

17.  bax  X  anx  X  3«  X  b^xy. 

18.  —  ^bz  X  —  xz  X  —  yz  X  agz. 

19.  '^cH  X  2xh  X  —  z^  X  —  bgz\ 

20.  —  c^x  X'dx  X  dp-  X  ay. 

21.  -  2e  X  -1y  XaXbx. 

22.  —  ^ax  X  'day  X  —  "Ic^y  X  —  xy. 

23.  ax^  X  ~  y"^  X  —  1  X  Sax  X  —  a^y, 

24.  m^x  X  —  n^x  X  —  mn^  X  —  tnK 

25.  —  abx  X  —  ay^  X  ax  X  d^x^. 

26.  ^^  X  qy'^  X  xy  X  —  ax. 

27.  abc  X  —  d^  X  ax  X  —1  X  Sax. 

28.  l/4flx  X  Sex  X  —  1/2^2:  X  —  4?/2  x  ^m, 

29.  —  Qmx  X  —  2n-x  X  l/6ac  X  —  l/5mK 

30.  ~  a  X  ^c  X  —  1  X  1/4  X  Sd^  X  4:xy  X  y. 

53.  Changing  the  Signs  of  an  Equation. — If  an  alge- 
braic expression  be  multiplied  by  —  1  its  signs  will  all  be 
reversed,  and,  of  course,  the  value  of  the  expression  will  be 


LAW  OF  ASSOCIATION.  55 

changed.  To  multiply  any  number  by  —  1  will  change  it 
into  the  corresponding  number  with  the  reverse  sign. 

If  both  members  of  an  equation  be  multiplied  by  —  1, 
the  value  of  each  member  will  be  changed,  but  their  equality 
will  not  be  destroyed.     (Why  not  ?) 

Hence  in  working  with  equations,  it  is  legitimate  to 
change  the  signs  at  any  stage'  of  the  operation,  provided 
that  the  sign  of  every  term,  simple  and  complex,  on  both 
sides  of  the  equation  be  changed. 

EXERCISE    XX. 

1.  ^  =  80  -  (a:  -  20)  +  (3a;  -  120).     Find  the  value 

of  X. 

2.  240  -{x  -\-  40)  =  20  +'  {bx  -  60)  -  (2a:  -  80). 
Find  the  value  of  x. 

3.  A  father  left  his  property  of  47,000  dollars  to  his 
four  children,  giving  the  eldest  four  times  what  he  gave 
the  youngest  less  as  much  as  he  gave  the  second,  to  the 
second  three  times  as  much  as  he  gave  the  youngest  less  as 
much  as  he  gave  the  third,  and  to  the  third  twice  as  much 
as  he  gave  the  youngest  •  less  2000  dollars.  What  did  he 
give  each  ? 

4.  Divide  81  into  five  parts  such  that  the  second  shall 
be  twice  the  first  less  eight,  the  third  shall  be  three  times 
the  first  less  the  second,  the  fourth  shall  be  four  times  the 
first  less  the  third,  and  the  fifth  shall  be  five  times  the  first 
less  the  fourth. 

54.  Distributive  Law  of  Multiplication  of  Integers. — 

Ex.  1.  (12  -f-  8)  X  4  =  20  X  4  =  80, 

and  12  .  4  4-  8  .  4  =  48  +  32  =  80. 

(12  -  8)  X  4  =  4  X  4  =  16, 


56 


MUL  TIPLICA  TIOK 


and  12  .  4  -  8  .  4  =  48  -  32  =  16. 

(-  12  +  8)  X  4  =  -  4  X  4  =  -  16, 
and  -  12  X  4  +  8  X  4  =  -  48  +  32  =  -  16. 

(-  12  -  8)  X  4  =  -  20  X  4  =  -  80, 

and  -  12  X  4  -  8  X  4  =  —  48  -  32  =  -  80. 

(12  +  8)  X  -  4  =  20  X  -  4  =  -  80, 

and  12  X  -  4  +  8  X  -  4  =  -  48  -  32  =  -  80. 


In  general, 

{^a  +  ^h)  X  *c 


*«  X  *^  +  *&  X  ^c. 


The  product  of  a  polynomial  and  a  monomial  factor  is 
the  Siim  of  the  products  of  its  several  terms  a7id  that  factor. 
This  is  known  as  the  Distributive  Lato  of  Multiplication. 

It  is  a  law  controlling  the  combination  of  multiplication 
with  addition  and  subtraction. 

The  trutli  of  the  Distributive  Law  may  be  shown  by 
the  following  conventional  arrangement  of  units  on  a  plane 
surface. 

If  a  vertical  and  a  horizontal  line  intersect  each  other  on 
a  plane,  they  will  divide  the 
plane  into  four  quarters,  or 
quadrants.  These  quadrants 
are  numbered  as  shown  in  Fig.  3. 
By  general  agreement,  units 
counted  to  the  right  of  the  ver- 
tical line,  whether  above  or  be- 
low the  horizontal  line,  are  re- 
garded as  positive;  while  those 
counted  to  the  left  of  the  vertical 
Also  uuits  counted  upward 


II 


III 


IV 


Fig.  3. 
line  are  regarded  as  negative, 
from  the  horizontal  line,  wliether  at  the  right  or  left  of  the 
vertical  line,  are  regarded  as  positive,  while  those  counted 


LAW  OF  ASSOCIATION 


57 


II 

o 

+h 

+b 

I 
o 

o 

o 

o  o  o  o 

o  o  o  o 

-a 

+a 

-a 

+a 

o  o  o  o 

o  o  o  o 

o 
o 
III 

-b 

-b 

o 
o 
IV 

downward  from  the  horizontal  line  are  regarded  as  nega- 
tive. 

The  quality  of  the  units  arranged  in  the  four  quadrants 
is  shown  in  Fig.  4,  the  units  being 
represented  by  the  small  circles. 

A  rectangle  of  units  in  any  quad- 
rant, as  shown  in  Fig.   5,  represents 
a  product  of  two  factors.     A  rect- 
angle in    the   first   quadrant    repre- 
sents a  positive  •  product,  since  it  is 
composed  of  two  positive  factors;  a 
rectangle  in  the  second  quadrant  rep- 
resents a  negative  product   (why  ?) ;  ^^^'  ^' 
a  rectangle  in  the  third  quadrant  represents  a  positive  prod- 
uct (why?);  and  a  rectangle  in 
the  fourth  quadrant  represents  a 
negative  product  (why  ?). 

To  represent  the  case  of  (a  -{- 
b)  X  c,  mark  a  -\-  h  units  in  a 
horizontal  row  in  the  first  quad- 
rant, and  repeat  the  row  c  times 
one  above  the  other  (Fig.  6). 
These  rows  represent  the  prod- 
uct of  a  -\-  h  and  c,  and  the 
Fro.  5.  vertical  dotted  line  between  the 

a  units  and   the   b  units  shows  that  this  product  is  the 
sum   of    the    two    products    ac 
and  be. 

To    represent    the    case     of 
{a  +  -b)  .  0   or    {a  —  b)  .  c.    ar- 
range  c   rows   of   a   units    each  — 
in  the  first  quadrant  and  c  rows 
of  ~b  units  each  in  the  second  ^^^-  ^• 

quadrant  (Pig.  7).     Each  complete  horizontal  row  will  be 
composed  of  a  +  ~^>  or  a  —  b  units,  and  the   c  rows  to- 


II 

~ab 

I 
+ab 

o  o  o  o  o 

o  o  o  o  o 

o  o  o  o  o 

+b 

+b 

o  oooo 

o  o  o  o  o 

o  o  o  o  o 

o  o  o  o  o 
-a 

0  o  o  o  o 

+a 

~a 

+a 

O  O  0  o  o 

o  O  O  0  o 

o  o  o  o  o 

~b 

~b 

o  o  o  o  o 

o  o  o  o  o 

o  o  o  o  o 

o  o  o  o  o 
+ah 
III 

O  O  O  O  0 

-ab 
IV 

{a  -\- b)  X  c  =  ac  -{- be 

ac  -\-    be 

oooo  .      o  o 

c  o  o  o  o  .      o  o 

oooo  .      o  o 

a  +      & 


-be 

+  «^ 

oo 

o  o  o  o 

c  oo 

o  o  o  o 

o  o 

oooo 

'b 

a 

58  MULTIPLICATION 

gether  represent  the  })roduct  of  {a  4-  ~h)  and  r,  or  (a—b)c. 
This  product  is  evidently  the  sum  of  the  two  products  ac 
and  —  be,  and  is  equal  to  ac  -\-  ( —  be), 
or  ac  —  be. 

The  two  expressions  ac  -\-  {—  be) 
and  ac  —  be  are  not  identical  in  mean- 
ing. The  former  represents  two  sets 
of  units,  one  positive  and  one  negative, 
and  indicates  that  they  are  to  be  com- 
FiG.  7.  bined  into  one ;  the  latter  represents 

one  set  of  units  and  indicates  that  it 
has  been  obtained  by  uniting  two  sets  of  units,  one  positive 
and  one  negative. 

Of  course  the  products  ac  and  —  be  tend  to  cancel  each 
other  wholly  or  in  part,  but  the  actual  cancellation  can  be 
expressed  only  when  the  products  are  numerals  or  similar 
terms  with  numeral  coefficients.  In  the  actual  illustration 
ac  represents  12  positive  units  and  —  be  6  negative  units, 
and  ac  —  be  represents  6  positive  units  obtained  by  cancel- 
ling 6  of  12  positive  units  by  6  negative  units.  Were  be>ac, 
the  result  of  the  cancellation  would  have  been  a  number  of 
negative  units  equal  to  the  arithmetical  difference  of  the 
two  products. 

So  long  as  ac  >  be,  the  expression  ac  —  be,  as  a  whole, 
is  positive,  and  denotes  that  the  operation  produces  a  sur- 
plusage of  the  kind  of  units  employed ;  and  when  ac  <  be, 
the  expression  ac  —  be,  as  a  whole,  is  negative  and  indicates 
that  the  operation  produces  a  deficiency  of  the  kind  of  units 
employed. 

EXERCISE  XXI. 

1.  Arrange  the  units  to  represent  the  case  (a-^b)  X  ~c 
and  show  that  it  equals  —  ac  —  be. 

2.  Arrange  the  units  to  represent  {~a  -\-  ~b)  X  c, 
or  {—  a  —  b)  X  c,  and  show  that  it  equals  ■  -  ac  —  be. 


LAW  OF  ASSOCIATION. 


59 


3.    Arrange   the    units   to   represent   {~a  +  ~h)  X  ~c, 
or  (—  ft  —  ^)  X  —  c,  and  show  that  it  equals  ac  -\-  be. 


Ex.  2.      (6  +  4)(3  +  2)  =  10  X  5  =1  50, 


and 


3  .  3  +  4  .  3  +  6  .  2  4-  4  .  2  =  18  +  12  +  12  +  8  =  50. 
In  general. 


ad 
o  o  o  o 


hd 
o  o  o 


+ 


To  represent  the  case 
{a  -\-  h){c  +  d),  arrange  c  -{-  d 
rows  containing  a  -\-  b  units 
each  in  the  first  quadrant 
(Fig.  8).  The  c  +  d  rows 
will  represent  the  product  of 
a  -\-  b  and  c  -\-  d.  This  prod- 
uct is  evidently  equal  to 

ac  -\-bc  -\-  ad  -\-  bd. 

The  product  of  a  poly- 
nomial and  a  polynomial  is 

the  sum  of  the  products  of  the  first  polynomial  and  each 
term  of  the  second. 


ac, 
o  o  o  o 
o  o  o  o 
o  o  o  o 
o  o  o  o 
o  o  o  o 


a     + 


Fig.  8. 


o 
o 

o  he 
o 
o 


55.  Extension  of  the  Application  of  the  Distributive 
Law. — The  distributive  law  of  multiplication  which  we 
have  demonstrated  for  integers  is  assumed  to  hold  for  all 
kinds  of  numbers  which  can  be  expressed  by  letters.  Hence 
the  last  two  definitions  hold  for  all  integral  algebraic  ex- 
pression in  which  the  multiplicand  is  an  integral  polyno- 
mial. 


60  MUL  TIP  Lie  A  TION. 

EXERCISE  XXII. 

I. 

1.    Arrange  the  units  to  represent  the  case 
(a  +  h){c  +  -d),  or  {a  +  h){c  -  d), 
and  show  that  it  equals 

ac  -\-  he  —  ad  ~  hd. 
Show  by  a  similar  arrangement  that 

2.  (a-\-h)('~c-\-d),  or  {a-\-h){  —  ('-\-d)  =  —ac—hc-\-ad-\-hd. 

3.  {a-\-h){~c-\-~d),  or  {a-\-h){—('—d)  —  —ac—bc—ad—hd. 

4.  (a-\-~b)(c-{-d),  or  (a—b){c-\-d)=ar—bc-\-ad—bd. 
6.   (a-\-~b){c-\--'d),  or  (a—b)(c—d)=ac—bc—ad-\-bd. 

6.  («H-~^)(~c+f?),  or  («— Z>)(  — ^-|-^)  =  — «6'-|-^c-|-«!<:/— M. 

II. 

7.  {a-{-~b){~c-{-~d),ov{a—b){—c—d)  =  —ac-\-bc—ad-\-bd. 

8.  (~a-h&)(c+fZ),   or  {—a-^b){c-]-d):=^—ac-\-bc—ad-\-bd. 

9.  (~«+Z>)(c+"rZ),  or  (—«H-^)(C  — <-/)=:— ft6'+^C+«^—M 

10.  (~a-\-b){~c-\-d),  or  (— ^-j-Z')(  — 6*-f-6Z)=«c— Z>6'— «fZ+^>r?. 

11.  {-a-{-b){-c-\--d),or{-a-\-b){-c-d)-^ae-bc-\-ad-bd. 
12    ("rt+~^)(6'+^?),  or  {—a  —  b){c-\-d)  =  —ac—bc—ad—bd. 

13.  (~«4-~^)(^+~^)jOr  (  —  <-«— /^)(6'—fZ)  =  —«6'—Z»c+r^r/+J^. 

14.  ("«4-~^)(~^+fO'Oi'(— <^'  — ^)(— ^+^0=^^^-r^^— «^^— *^- 

15.  (~a-{-~b)(-c-\-^d),OY{—a  —  b){—c—d)=ac-{-bc-{-ad-^bd. 

"N^ote  that  the  numbers  in  the  adjacent  quadrants  tend 
to  cancel  each  other,  while  those  in  the  opposite  quadrants 
tend  to  augment  each  other.  The  expression  finally  ob- 
tained will  be  positive  or  negative  according  as  the  sum  of 


LAW  OF  ASSOCIATION.  61 

the  units  in  the  first  and  third  quadrants  is  greater  or 
less  than  the  sum  of  those  in  the  second  and  fourth  quad- 
rants. 

56.  Arrangement  of  Terms  according  to  the  Powers 
of  a  Letter. — A  polynomial  is  said  to  be  arranged  accord- 
to  the  powers  of  some  letter  when  the  exponents  of  that 
letter  either  ascend  or  descend  in  magnitude  in  regular 
order.  Thus,  ba  —  iSbx  -\-  'dcx^  —  4:a^x^  is  arranged  accord- 
ing to  the  ascending  powers  of  x;  and  3^^  —  ^ax"^  -{-  ex  —  7 
is  arranged  according  to  the  descending  powers  of  x. 

57.  Multiplication  of  Polynomials. — (a)  To  multiply  a 
polynomial  hy  a  monomial,  multiply  each  term  of  the  poly- 
nomial by  the  monomial,  and  tv7'ite  the  result  as  a  poly- 
nomial reduced  to  its  simplest  form. 

EXERCISE   XXIII. 

Multiply  together: 

I. 

1.  3xy  -\-  4:yz  and   —  VZxyz. 

2.  ab  —  be  and  a^be^. 

3.  -•  X  —  y  —  z  and  —  ^x. 

4.  a^  —  b'^  -\-  c^  and  abc. 

5.  —  ab  -\-  be  —  ca  and  —  abc. 

6.  -2a^  ~  4:ab^  and  -  7d^b\ 

7.  5x^y  —  Qxy'^  +  Sx^y^  and  dxy. 

8.  ~  7x^y  —  bxy^  and  —  %x^y^. 

9.  —  bxyh  -{-  dxyz^  —  Sx^yz  and  xyz, 

10.  ix^yh^  —  Sxyz  and  —  12x^yz^.  , 

11.  —  13a;y^  —  Ibx^y  and  —  7c(^y'^. 


62  MULTIPLICATION. 

II. 

12.  ^xyz  —  lOx'^yz^  and  —  xyz, 

13.  ahc  —  a%c  —  ab'^c  and  —  «^c. 

14.  —  (v^bc  +  ^^6'r«  —  c^ab  and  —  0^6. 
Find  the  product  of 

15.  2rt.  —  'db-\-  4:0  and  —  3/2«. 

16.  3x  —  2//  —  4  and   —  6/Qx. 

17.  2/3r«  —  J/6^  —  c  and  d/Sax. 

18.  6/7«V  _  3/2f/.r3  ^nd  -  7/Sa^x. 

19.  -  5/3«V  and  -  3/2«2  +  «a:  -  3/5^1 

20.  -  7/-Zxy  and   -  3.^2  +  2/7x1/ . 

21.  -  3/22-y  and  -  l/3i;2  +  2if. 

22.  -  4/7.<//3  and  7/4:X^  -  4:/7f. 

(b)  To  multiply  a  ptolynowial  by  a  2^olynomial,  rmiUiply 
the  first  polynomial  by  each  term  of  the  second,  and  add 
the  partial  products  thiis  obtamed. 

In  multiplying  polynomials  it  is  convenient  to  arrange 
the  terms  of  both  factors  in  the  same  order  according  to  the 
powers  of  some  letter,  to  write  the  multiplier  under  the 
multiplicand,  and  to  place  like  terms  of  the  partial  products 
in  columns. 

e.g.  (1)  Multiply  4:X -\- 'd  ^  bx^  -  ^x^  by  4  -  Qx^  -  5.r. 

Arrange  both  multiplicand  and  multiplier  according  to 
the  ascending  powers  of  x. 

3+    4a:  +    5:^2  _     g^ 
4  -    5a;  -    6:^2 


12  +  I62;  +  20.c2  -  Ux? 

-  I6x  -  %W  -  2bx^  +  3(^-4 

-  18a;2  -  24a;3  -  30.^^  -f  36a-5 

12  -f      X-  Wx^  -  7^3^  +  36a:5 


LAW  OF  ASSOCIATION.  63 

(2)  Multiply  l-i-2x  +  x^  -  S^  hjx^-2-  2x. 
Arrange  according  to  the  descending  powers  of  x. 

x^  -dx^-^2x-\-l 

a^-2x  -2 


x^  -  dx^  +  2x^  +    x^ 

-   2jf                        H-   QX^   -   4:X^   - 

-2x 

-  2x'            +  Qx^  - 

■4:X- 

■  2 

X^  -  5a;5              4-  7x^  +  2x^  - 

-Qx- 

-2 

EXERCISE  XXIV. 

\Iu 

Itiply  together : 

I. 

1. 

X  -^1  and  X  —  1. 

2. 

x^  -\-  xy  -\-  y^  and  xy. 

3. 

a?  -?>x^-\-x-  ^  and  -  ?>x^. 

4. 

:c2  +  a;  +  1  and  cc2  -  1. 

5. 

x^  +  2a;  +  3  and  x^  -x^\. 

6. 

X?  -hx^^  and  x?  +  5:r  +  6. 

7. 

X?  -\-  xy  -{-  y^  and  x  —  y. 

8.  x^  —  xy  -\-  y^  and  x  -\-  y. 

9.  x^  +  xy  -\-  y^  and  x^  —  xy  -{■  y^. 

10.  .'2;^  +  3x^  H-  Sa;  +  1  and  x^  +  2a;  +  1. 

11.  3(a;  -  4)  =  361  +  8(2a;  -  12)  -  5(4a;  +  40). 
Clear  of  parentheses  and  find  the  value  of  x. 

12.  A  man  bought  three  houses.  He  paid  for  the  sec- 
ond 8000  dollars  less  than  three  times  as  much  as  he  paid 
for  the  first,  and  for  the  third  five  times  what  he  paid  for 
the  first  less  the  cost  of  the  second.     Five  times  the  cost  of 


64  MULTIPLICATION. 

the  first  minus  the  cost  of  the  second  is  equal  to  192,000 
dollars  minus  three  times  the  cost  of  the  third.  What  was 
the  cost  of  each  house  ? 

13.  A  man  started  to  give  50  cents  apiece  to  some  beg- 
gars and  found  he  had  not  money  enough  within  7  cents. 
He  then  gave  them  45  cents  apiece  and  had  18  cents  left. 
How  many  beggars  were  there  ? 

II. 
Multiply  together: 

14.  x^  —  2ax^  +  2 A  —  da^  and  x^  —  Sax  +  2a^. 

15.  ^  —  ax^  —  2  A  -j-  a^  and  x'^  -\-  ax  —  a^. 

16.  x^  -\-  iix^y  +  Qx^y'^  +  4.^'?/^  +  y^  and  x'^  —  2xy  -\-  y^, 

17.  X  —  a,  X  +  ci,  and  x^  -\-  a^. 

18.  X  —  a,  X  -{-  b,  and  x  —  c. 

19.  \-\-x-\-Q^,\  —  x-\-  o;^,  and  1  —  x  -\-  x^. 

20.  a  —  i,  a  -\-  b,  a^  —  ah  -\-  W,  and  c?  ^  ab  -\-  W'. 

21.  ^x?  4-  Vlxy  +  \^y^  and  3cc  —  4?/. 

22.  25«V  -  IhaWxy'^  +  UHf  and  5 A  +  Wf. 

23.  16«V  +.20rt//li-^''^  +  25^ V  and  ^az^  -  Wx. 

24.  A  man  bought  three  horses.  He  paid  50  dollars 
less  than  twice  as  much  for  the  second  as  for  the  first,  and 
for  the  third  three  times  the  cost  of  the  first  less  the  cost 
of  the  second.  Seven  times  the  cost  of  the  first  minus 
twice  the  cost  of  the  second  is  equal  to  1700  dollars  minus 
twice  the  cost  of  the  third.     What  was  the  cost  of  each  ? 

25.  A  man  gave  some  beggars  30  cents  apiece  and  had 
12  cents  left.  He  found  that  he  needed  four  cents  more 
to  enable  him  to  give  them  32  cents  apiece.  How  many 
beggars  were  there  ? 

68.  Multiplication  by  Detached  Coefficients.  —  When 
two  expressions  contain  one  and  the  same  letter  and  both 


LAW  OF  ASSOCIATION.  Q)^ 

are  arranged  according  to  the  ascending  or  descending 
powers  of  that  letter,  much  labor  of  multiplication  can  be 
saved  by  writing  down  the  coefficients  only. 

Thus,  to  multiply  q-?  —  5a*  +  6  by  x^  -\-  bx-\-  Q,  we  write 

1-5+6 
1  +  5+  6 
1-5+6 

5-25  +  30 

6-30  +  36 


1  +  0-13+    0  +  36 

The  highest  power  of  x  in  the  result  is  x'^,  and  the  rest 
follow  in  order.     Hence  the  required  product  is 

^4  _^  ox^  -  l^x^  +  Oa;  +  36, 

or  a;^  -  13^2  _^  35^ 

When  some  of  the  powers  of  the  letter  are  wanting,  the 
coefficients  must  be  written  down  as  zeros  in  their  jiroper 
places.  Thus,  to  multiply  x^  +  '6x^  +  3a;  +  1  by  a;^  +  'Zx^ 
+  1,  we  write 

1+0+3+  1 

1  +  2+  0  +  1 

1+0+3+  3+1 

2  +  0+  6  +  6  +  2 

0+  0+0+0+0 

1+0+3+3+1 


1  +  2  +  3  +  10  +  7  +  5  +  3  +  1 
Hence  the  product  is 

xi  +  2a;«  +  3a;5  +  10.^:*  +  7.^3  +  bx^  +  3a;  +  1. 

The  method  illustrated  above  is  known  as  the  method  of 
detached  coefficients. 


QQ  MULTIPLICA  TIOK 

EXERCISE  XXV. 

Do  the  following  multiplications  by  the  method  of  de- 
tached coefficients. 
Multiply : 

I. 

1.  3a:2  -  a:  4-  2  by  ^x^  +  'Zx  -  2. 

2.  ic*  -  2a:2  +  a:  -  3  by  2;*  4-  2-3  -  :^^  -  3. 

3.  ^  -5x^  +  1  by  %x^  +  5a;  4-  1. 

4.  ^x^  -3x^-{-x-2  by  x^  -2x^-x-^% 

5.  1  -  2x  +  a;2  by  1  +  2x  +  3x^  +  4:X^  +  6x\ 

6.  1  +  'Zx  +  3a;=^  +  4^-3  +  5a;4  +  Gx^  by  1  -  2.C  4-  or^. 

7.  1  -  2x  4-  'da^  by  1  4-  3:z;  -  52;^. 

8.  Z  -\-  Sx  —  2x^  by  2  —  3a;  4"  ^^^« 

9.  x^  -2x^-{-x-\-l  by  .^2  4-  1. 
10.  x''  -2x^  +  3  by  2x'  -  xK 

II. 
Examples  1-10  of  Exercise  XXIV. 

69.  Degree  of  an  Integral  Expression. — 'the  degree  of 
an  integral  term  in  any  letter  is  the  number  of  times  that 
letter  is  contained  as  a  factor  in  the  term,  and  is  equal  to 
the  exponent  of  the  letter. 

The  degree  of  an  integral  term  in  two  or  more  letters  is 
the  number  of  times  all  together  that  these  letters  occur  as 
factors  in  the  term,  and  is  equal  to  the  sum  of  the  expo- 
nents of  the  letters  in  the  term. 

The  degree  of  a  term  in  any  letter  or  letters  is  often 
called  the  dime?isio7i  of  the  term  in  that  letter  or  those 
letters. 

The  degree  of  any  integral  algebraic  expression  in  any 


LAW  OF  ASSOCIATION.  67 

letter  or  letters  is  the  degree  of  the  term  in  it  which  is  of 
the  highest  dimensions  in  that  letter  or  those  letters. 

e.g.  The  term  5a^b^x^  is  of  the  fifth  degree  in  x,  of  the 
nintli  degree  in  bx,  and  of  the  twelfth  degree  in  abx. 

The  expression  ba^x'^  +  Qa^a^  —  llax^  is  of  the  sixth 
degree  in  x  and  of  the  seventh  degree  in  ax. 

It  will  be  noticed  that  in  the  last  example  every  term  is 
of  the  same  degree  in  ax.  When  all  the  terms  of  an  ex- 
pression are  of  the  same  degree  in  any  letters,  the  expression 
is  said  to  be  homoge^ieous  in  these  letters. 

60.  Product  of  Homogeneous  Expressions. — The  prod- 
uct of  tivo  homogeneous  expressions  must  be  homogene- 
ous.— For  each  the  terms  of  the  product  is  obtained  by 
multiplying  some  one  term  of  the  multiplicand  by  some  one 
term  of  the  multiplier,  and  the  number  of  dimensions  of 
the  product  of  two  terms  is  clearly  the  sum  of  the  number 
of  dimensions  of  the  separate  terms.  Hence,  if  all  the 
terms  of  the  multiplicand  are  of  the  same  degree,  and  all 
the  terms  of  the  multiplier  are  also  of  the  same  degree,  it 
follows  that  all  the  terms  of  the  product  must  be  of  the 
same  degree. 

It  also  follows  from  the  above  consideration  that  the 
degree  of  the  product  is  the  sum  of  the  degrees  of  the  fac- 
tors. 

When  the  two  factors  to  be  multiplied  are  homogene- 
ous, there  must  be  some  error  if  the  products  obtained  are 
not  homogeneous. 

61.  Highest  and  Lowest  Terms  of  a  Product. — It  is  im- 
portant to  notice  that,  in  the  product  of  two  algebraic  ex- 
pressions, the  term  which  is  of  the  highest  degree  in  any 
particular  letter  is  the  product  of'  the  terms  in  the  factors 
which  are  of  the  highest  degree  in  that  letter,  and  the  term 
which  is  of  the  lowest  degree  in  that  letter  is  the  product 
of  the  terms  which  are  of  the  lowest  degree  in  that  letter  in 


68  MULTIPLICATION. 

the  factors.  Thus  there  can  be  obtaiued  only  one  highest- 
degree  term  and  one  lowest-degree  term. 

62.  Complete  and  Incomplete  Integral  Expressions. — It 

is  also  important  to  notice  that  if  each  factor  in  mnltiplica- 
tion  is  complete  in  any  letter,  that  is,  contains  every  degree 
of  that  letter  from  the  highest  one  given  down  to  zero,  the 
product  will  be  complete  in  that  letter. 

Thus  the  product  oi  x^  -\-  x^  -\-  x^  -\-  1  and  x'^  -\-  x  -\- 1 

is  x^  -f  2r^  +  ^x^  +  ^x^  +  3a;2  +  2.t  +  1. 

If  an  expression  is  incomplete  in  any  letter  it  may  be 
completed  by  filling  in  the  blank  spaces  with  terms  of  the 
proper  degree  having  zero  as  their  coefficients.  Thus 
x^  -f  a:'-^  -f  1  may  be  written  x^  +  Ox^  +  Ox^  -\- x^ -\- Ox -^  1. 


CHAPTER  VII. 

DIVISION  OP  INTEGRAL  ALGEBRAIC 
EXPRESSIONS. 

63.  Definition  of  Division. — Division  is  the  inverse  of 
multiplication,  or  the  process  of  undoing  multiplication. 
In  multiplication  two  factors  are  given  and  their  product  is 
required.  In  division  the  product  and  one  of  the  factors 
are  given  and  the  other  factor  is  required. 

The  product  of  the  two  factors  is  called  the  dividend, 
the  given  factor  the  divisor,  and  the  required  factor  the 
quotient. 

Since  the  dividend  is  the  product  of  the  divisor  and 
quotient,  we  may  prove  our  division  by  multiplying  together 
the  divisor  and  quotient  to  see  if  their  product  agrees  with 
the  dividend. 

64.  Division  of  Monomials. — The  rules  for  division  are 
obtained  by  studying  the  corresponding  cases  of  multiplica- 
tion. 

Take  the  following  cases  of  the  multiplication  of 
monomials : 


Note:  1°.  That  the  sign  of  one  factor  is  +  when  the 
signs  of  the  product  and  of  the  other  factor  are  alike,  and 


70 


DIVISION. 


—  when  the  signs  of  the  product  and  of  the  other  factor 
are  unlike. 

2°.  That  the  coefficient  of  one  factor  is  the  quotient 
obtained  by  dividing  the  coefficient  of  the  product  by  the 
coefficient  of  the  other  factor. 

3°.  That  the  exponent  of  any  letter  in  one  factor  is  the 
difference  between  its  exponent  in  the  product  and  in  the 
other  factor,  and  that  when  this  difference  is  zero  the  letter 
does  not  appear  in  the  other  factor.  When  any  letter  which 
appears  in  the  product  does  not  appear  in  one  factor,  its 
exponent  in  that  factor  is  to  be  regarded  as  zero. 

From  these  observations  we  obtain  the  following  rule 
for  the  division  of  a  monomial  by  a  monomial : 

Divide  the  coefficient  of  the  dividend  hy  that  of  the  divi- 
sor for  the  coefficient  of  the  quotient,  subtract  the  expoiient 
of  each  letter  in  the  divisor  from  its  exponent  in  the  dividend 
for  its  exponent  in  the  quotient,  and  place  before  the  term 
in  the  quotient  the  plus  sign  when  the  sights  of  the  divisor 
and  dividend  are  alihe,  and  the  minus  sig7i  when  the  signs 
of  the  divisor  and  dividend  are  tinlike. 


EXERCISE  XXVI. 


Divide : 


1.     20a^y  by  4a;l 

3.     5^a*b^c  by  6aH^c., 

5.     blaxh  by  —  3azx^. 


2.     21a^  by  7b. 

4,     4:9a^yh  by  7xyh, 

6.     -  132a^yh  by  12^2^. 


II. 


7.     —  Sbx^yh"^  by  - 

9.     l/5.cy  by  l/lOa^y.  10. 

11.     -  2/3ay  by  -  5/6a^y.      12. 


-'27a^c^\)j-Sabc^. 
l/^a^b"  by  -  \/Vlab^, 
-  Oary^s  by  %/3xt'. 


Division.  tl 

Multiply : 

I. 

13.  b{x-[-  ijfz  by  3(a;  +  i/)V. 

14.  13(rt  -  Ifx  by  -  3(^/  -  VfT?. 

16.     -  5c(a  +  Z>)4a:y  by  U{a  +  ^)3r2. 

II. 

16.  -  na%{c  -  d)y^  by  -  ^a})\c  -  dfx. 
Divide: 

17.  45(rt  +  ifx^  by  9(«  +  ^).r2. 

18.  Q'dac\b  -  d^xf  by  -  7c(^>  -  ^)2a:^. 

19.  -  ^'lc^d{h  +  c)2:^2  by  _  3c2(j  _|_  c)^^ 

Simplify: 

I. 

20.  (V'h'^C,  X    (-  8r«3J4^,5)  _^   _  4^6j6p4^ 

21.  -  ^xhf  X  (-  12.<?/«)  -^  -  4:ry. 

22.  260  -  3(a;  -  2)  =  14  +  4(a;  +  3)  -  12:^2  _^  4^^ 

23.  Divide  180  into  two  parts  such  that  80  minus  three 
times  the  sum  of  the  smaller  part  and  12  shall  be  equal  to 
the  larger  part  minus  8  less  than  five  times  the  smaller 
part. 

65.  Division  of  Polynomials.— «.  We  have  seen  in  multi- 
plication that,  when  one  of  the  factors  is  a  monomial  and 
the  other  a  polynomial,  the  product  will  be  a  polynomial, 
and  that  this  product  is  obtained  by  multiplying  each  term 
of  the  polynomial*  factor  by  the  monomial  factor.  Hence  in 
division,  when  tbe  dividond  is  a  polynomial  and  the  divisor 
is  a  monomial,  the  quotient  will  be  a  polynomial,  and  this 
quotient  will  be  obtained  by  dividing  each  term  of  the  divi- 


72  DIVISION. 

dend  by  the  divisor.     Of  course,  the  law  of  signs  must  be 
carefully  observed. 

EXERCISE  XXVII. 

Divide : 

I. 

1.  a^'jf  -\-  a^y^  +  ^y'^  ^1  ^'i^* 

2.  a^h  —  a%^  +  a^V^  by  aH. 

3.  -  2a^h  +  ^a%^  -  2ab^  by  -  2ah. 

4.  24:a^y^  +  lOSx^y^  +  Slxyf  by  'dxf, 

5.  a'b^  -  Q/'25a'b^  -  2/5a^b^  by  Q/6ai\ 

II. 

6.  Ua^b^  +  28a3J*  by  -  7a^\ 

7.  15a;y  —  182:^,1?/^  +  Mx^y^  by  3a;«/. 

8.  -  3«2  -|.  9/2<^^)  -  6ac  by  -  3/2«. 

9.  -  6/2x^  +  5/3a:?/  +  lO/'Sx  by  -  5/6a:. 
10.     1/4  A  —  l/16«Z»a;  —  3/8«c:c  by  3 /Sax. 

66.  J.  In  multiplication,  we  have  seen  that,  when  each 
factor  is  a  polynomial,  their  product  is  the  sum  of  the  par- 
tial product  obtained  by  multiplying  the  whole  multiplicand 
by  each  term  of  the  multiplier.  In  this  case  the  product  is 
a  polynomial. 

Hence  in  division,  when  the  divisor  is  a  polynomial,  we 
obtain  a  set  of  partial  subtrahends  by  multiplying  the  whole 
divisor  (the  multiplicand)  by  each  term  of  the  quotient,  as 
it  is  found.  These  partial  subtrahends  are  subtracted  in 
succession  from  the  dividend.  The  operation  is  continued 
until  there  is  no  remainder,  or,  in  case  tlie  divisor  is  not  an 
aliquot  part  of  the  dividend,  until  the  remainder  is  of  a 
lower  degree  than  the  divisor. 


DIVI8I0N.  73 

The  method  of  procedure  in  division  will  be  readily 
understood  by  examining  a  case  in  multiplication  of  poly- 
nomials, and  the  corresponding  case  in  division. 

e.g.  x^  —  ^x^  -\-    4a^ 

3a;2  _  2a;  -    7 


3ic«  -  9ar*  -f  12a;* 

-  2a;S  +    6a;4  -    8a;« 

-    7a;*  +  21a:3  -  28a;2 


3a;«  —  ll.r''  +  II.t*  +  IBa;^  -  28a;2 

Note  that  the  first  term  of  the  first  partial  product  is 
also  the  first  term  of  the  complete  product,  and  that  it  is 
the  product  of  the  first  term  of  the  multiplier  and  multipli- 
cand. Hence,  in  dividing  the  product  by  one  factor,  the 
first  term  of  the  other  factor  will  be  the  quotient  obtained 
by  dividing  the  first  term  t)f  the  dividend  by  the  first  term 
of  the  divisor,  and  the  first  partial  subtrahend  (partial  prod- 
uct) will  be  obtained  by  multiplying  the  whole  divisor  by 
this  first  term  of  the  quotient.     Thus : 

3^^f.  _  11^5  _|_  11^4  _|.  13^3  _  14^  I  ^4  _  3^3  _|_  4^ 

3.^r,  _    9^    I    i2a;4  ^y^ 


-   2x'  -      x'+  13x^  -  Ux^ 

Note  again  that  the  first  term  of  the  remainder  just  ob- 
tained is  also  the  first  term  of  the  second  partial  product  in 
the  corresponding  multiplication,  and  that  it  is  the  product 
of  the  first  term  of  the  factor  used  as  a  divisor  and  the 
second  term  of  the  other  factor  or  quotient.  Hence  in  di- 
vision the  second  term  of  the  quotient  will  be  obtained  by 
dividing  the  first  term  of  the  first  remainder  by  the  first 
term  of   the  divisor,    and  the  second  partial  subtrahend 


H  DIVI8I0K. 

(partial  product)  will  be  obtained  by  multiplying  the  whole 
divisor  by  this  second  term  of  the  quotient.     Thus : 

3»6  _  11^:5  +  11:^4  _^  13.^3  _  28:^2  |  «;'  -  3x^  +  ^x^ 

3^6  _    9^5  _^  i2a:4  32;2  -  2a;  -  7 


--2^^ 

— 

^4  _|_  13^3  _ 

-28a;2 

-    2^5 

+ 

6:c*-     ^yf 

— 

7£c^  +  21a:3  - 

-  28a;2 

— 

1x^  +  21a;3  - 

-28a;2 

Note  as  before  that  the  first  term  of  the  second  remain- 
der is  the  same  as  the  first  term  of  the  third  partial  product, 
and  that  it  is  the  product  of  the  first  term  of  the  factor 
used  as  the  divisor  and  the  third  term  of  the  other  factor 
or  quotient.  Hence  in  division  the  third  term  of  the 
quotient  will  be  obtained  by  dividing  the  first  term  of  the 
second  remainder  by  the  first  term  of  the  divisor,  and  the 
third  partial  subtrahend  (partial  product)  will  be  obtained 
by  multiplying  the  whole  divisor  by  this  third  term  of  the 
quotient. 

Should  there  be  another  remainder,  the  next  term  of  the 
quotient  will  be  obtained  in  a  similar  way. 

Use  the  second  factor  in  the  preceding  case  of  multipli- 
cation as  a  divisor,  and  go  through  the  work  in  the  same 
way,  and  note  the  same  points. 

Also  go  through  the  same  case,  arranging  the  terms  of 
divisor  and  dividend  according  to  the  ascending  powers  of  x. 

It  is  customary  to  bring  down  only  one  term  at  a  time, 
and,  in  case  the  dividend  is  not  exactly  divisible  by  the  di- 
visor, to  express  the  remainder  in  the  form  of  a  fraction  as 
in  arithmetic. 

When  some  of  the  powers  of  the  letter  according  to 
which  the  terms  are  arranged  are  wanting,  their  places  may 


DIVISION,  75 

be  supplied  by  terms  with  zero  coefficients.    Thus,  suppose 
the  dividend  to  be  ^®  —  27 :   it  may  be  written 

a;6  ^  Qx'o  j^  0:^4  -H  Qx^  +  Oz^  -^  Ox  -  27. 

This  is  not  absolutely  necessary,  but  will  be  found  con- 
venient. 

The  rule  for  the  division  of  a  polynomial  by  a  polynomial 
may  be  stated  as  follows : 

Arrange  the  terms  of  the  divisor  and  dividend  simi- 
larly;  divide  the  first  term  of  the  dividend  by  the  first  term 
of  the  divisor  for  the  first  term  of  the  quotient,  and  multi- 
ply the  divisor  hy  this  term  for  the  first  partial  subtrahend ; 
divide  the  first  term  of  the  remainder  by  the  first  term  of 
the  divisor  for  the  second  term  of  the  quotient,  and  multiply 
the  divisor  by  this  term  for  the  second  partial  subtrahend  ; 
and  continue  the  process  until  there  is  no  remainder,  or 
until  the  first  term  of  the  remainder  does  not  contain  the 
first  term  of  the  divisor. 


Div 

EXERCISE  XXVIII 

ide: 

1. 

I. 
x^  -x-^hy  x  +  ^. 

2. 

x^  —  4:X  —  21  by  a;  —  7. 

3. 

x^  —  12a;  +  35  by  a;  —  5. 

4. 

2x^-x-  iJhy  2x  +  3. 

6. 

Qx^  -rdx-\-Q  by  'dx  -  2. 

6. 

nx^  +  11a;  -  56  by  4a;  -  7. 

7. 

16a;2  -  24a;  +  9  by  4a;  -  3. 

8. 

25a;2  -  16  by  5a-  --  4. 

9. 

49a-2  -4-  70a;  +  25  by  7a;  +  5. 

LO^ 

x^  -  y^hy  X-  y. 

76 


DIVISION. 


11.  x^  -f-  y^  by  x^  —  xy  -\-  y"^, 

12.  27«V  -  64^3  by  ^ax  -  4&. 


II. 

13.  8aV  -  27c«^9  by  4«V  +  Qa^^(^x^  +  9c^J«. 

14.  14a:4  +  45rc3«/  +  78a;2?/2  +  45a:?/3  +  14?/'^   by    %x^  + 

7/2 


+  3  by  ^2  _  3^,  _^  2. 


5^;?/  +  ly^ 

16.  a:^  -  ^^  +  9a:«  -  6.^2 

16.  x^  —  4:X^  +  da^  +  3.T=^  -  3a;  +  2  by  a;2  _  a;  —  2. 

17.  x^  —  x^y  -{-  x?y^  —  Q(f  —  y^hj  x^  —  X  —  y. 

18.  i^^  +  x^y  —  x?y^  +  x^  —  ^xy"^  -\-  y^  by  x'^-\-xy  — 

19.  a;5  -  2^;*  —  4a:3  _^  19^2  _  3^^  +  15  by  ^ 

20.  2a;3  -  8a;  4  rc^  +  12  -  7^  by  a;2  +  2  -  3a;. 

21.  14«^  -  45«35  +  78«2^2  _  45^j3  _^  ^4^4  ^y   ^a^  _ 

Find  the  remainder  in  each  of  the  following  examples: 


7a; +  5. 


23. 
24. 
25 
26. 
27. 


30. 


a;3  -  6a;2  +  11^  +  2 
x^  -  6a;2  +  12a;  -  17 
2a;3  4-  5:^:2  _  4a;  _  7 
3^3  _  7a;  -  9 
4a;3  +  7a;2  -  3a;  -  33 
27a;3  +  9a;2  -  3a;  -  5 
16a;3  -  19  +  39a;  -  46a;2 
8a;  -  8a;2  +  5a;3  +  7 
21«3-27a  +  15  -26«2 


divided  by  a;  —  2. 

'  a;- 3. 

'  a; +  2. 

'  x-\-l. 

'  4.x  ~  5. 

*  3a; -2. 

'  8a-  -  3. 

'  5a-  -  3. 

'  3a-  9. 


DIVISION.  77 

II. 

SI.  30a;*  +  lla;^  -  82a;2  -  5a;  +  3  divided  by  2a;  -  4 
+  3a;2. 

32.  6a;  -  5a;3  +  12a;*  +  20  -  33a;2  divided  by  a;  +  4a;2 
-5. 

33.  30a;  4-  9  -  IW  +  28a;*  —  35a;2  divided  by  4.x^— 
Vdx  +  6. 

Divide : 

34.  2a;2  +  7/6a;  +  1/6  by  2a;  +  1/2. 

35.  l/3a;3  +  17/6a;2  _  5/4^;  +  9/4  by  l/3a?  +  3. 

36.  1  by  1  +  a;. 

37.  1  +  a;  by  1  —  a;. 

38.  4(a;  -  yf  -  16(a;  -•  yf  —  8(a;  -  y)^  -  {x  -  y)  by 
2(a;  -  yf  +  4(a;  -  ^)  +  1. 

The  division  of  a  polynomial  by  a  polynomial  may  be 
indicated  by  writing  the  divisor  after  the  dividend,  each 
enclosed  within  a  parenthesis,  with  the  sign  of  division  be- 
tween.    Thus,  (a;2  -f  12a;  +  35)  -^  {x  +  7)  =  ^-  +  5. 

67.  To  Free  an  Equation  from  Expressions  of  Division. 

—  Since  multiplication  by  any  quantity  neutralizes  the 
effect  of  division  by  the  same  quantity,  and  since  to  multi- 
ply both  members  of  an  equation  by  the  same  quantity  does 
not  destroy  their  equality,  an  equation  may  be  freed  from 
an  expression  of  division  in  either  member  by  multiplying 
both  members  by  the  indicated  divisor. 

e.g.  4  +  (5a;2  -  40)  -^  {x  -  3)  =  5a;, 

4(a;  -  3)  +  bx^  _  40  =  Sa;^  _  i^y.^ 
or  4a;  -  12  +  bx^  -  40  =  Sar^  -  16a;, 

or  4a;  +  15a;  -}-  bx^  —  5ar^  =  52, 


78  DIVISION. 

or  19a;  =  52, 

The  above  example  might  have  been  written 

,,    5^2-40       . 
X—  d 

N.B. — In  clearing  an  equation  of  a  fraction  it  must  be 
borne  in  mind  that  the  bar  of  the  fraction  is  a  sign  of  ag- 
gregation, and  requires  a  change  of  sign  when  there  is  a 
minus  sign  before  the  fraction. 

EXERCISE  XXIX. 

Free  the  following  equations  of  expressions  of  division 
or  fractions : 

I. 

1.    — -_ -,.77—  =  5a;  —  4. 

(30a;  -  60)  ^  (7a;  -  16)  =  Qx  -  3. 
3.     6a;  +  7  -  5(2a;  -  2)  -^  (7a;  -  16)  =  3(2a;  +  1). 

,.       35(;?;  -  5)       „ 

5.  ,7a;  -  6  -  — ^ — --^  =  7a;. 

bx  —  101 

6.  A  woman  buys  eggs  at  18  cents  a  dozen.  Had  she 
bought  five  dozen  more  for  the  same  money,  the  eggs  would 
have  cost  her  2^  cents  a  dozen  less.  How  many  dozen  did 
she  buy  ? 

7.  A  man  bought  some  sheep  "at  three  dollars  a  head. 
Had  he  bought  two  less  for  the  same  money,  they  would 
have  cost  him  one  dollar  more  a  head.  How  many  did  he 
buy? 


X  - 

-  7a; 

6a;  + 

1  - 

6a;  + 

7  - 

6a; + 

13- 

SYNTHETIC  DIVISION.  79 

68.  Division  by  Detached  Coefficients. — It  is  evident 
if  the  dividend  and  divisor  are  both  homogeneous,  the  de- 
gree of  the  quotient  will  be  that  of  the  dividend  minus  that 
of  the  divisor. 

Also  if  the  dividend  and  divisor  are  complete  in  any 
letter,  the  quotient  will  also  be  complete  in  that  letter. 

In  finding  the  quotient  of  two  integral  algebraic  expres- 
sions which  are  arranged  in  the  same  order  according  to 
the  powers  of  some  letter,  much  labor  may  be  saved  by  the 
method  of.  detached  coefficients. 

e.g.  Divide  l^x^  +  ^x^  -  l^x^  +  4:3?  +  l^x^  +  16a:  -24 
by  4x^  +  2x^  -  4. 

12  +  6  -  16  +    4  -h  12  +  16  -  24  I  4  +  2  +  0  -  4 

12  4-6+0- 12  3  +  0-4  +  6 


0-16  +  16  +  12 

0+    0+    0+    0 

-  16  +  16  +  12  +  16 

-  16  -    8+    0  +  16 

24+12+    0- 

-24 

24  +  12  +    0  - 

-  24 

The  required  quotient  is  ^x^  —  4x  +  6. 
EXERCISE  XXX. 
II. 
Exercise  XXVIII,  Examples  15-20. 

SYNTHETIC    DIVISION. 

N.B. — This  section  may  be  omitted;  but  if  mastered,  it 
will  lead  to  an  immense  saving  of  labor  in  the  end,  even  in 
Elementary  Algebra, 


80 


DIVISION. 


69.  Synthetic  Multiplication. — In  the  first  place  let  us 
examine  some  cases  of  what  may  be  called  synthetic  multi- 
plication ;  that  is,  multiplication  of  complete  integral  al- 
gebraic expressions  in  which  the  coefficients  of  the  several 
powers  of  the  letter  are  built  up  one  after  another.  This 
is  effected  by  a  kind  of  cross-multiplication,  with  which 
one  may  be  made  familiar  by  a  little  practice. 

e.g.  1.  Multiply  pa^  -j-  qx^  -\-  rx  -\-  s  by  ax^  -\-  bx  -\-  c, 

px?  -j-  qx^  ■\-  rx-\-  s 

ax^  •{-hx  -{•  c 


Ax'  +  Bx^  +  Ca^  +  Dx^  -\-  Ex -]-  F. 

The  first  coefficient  of  the  product  is  formed  of  the  first 
coefficients  of  the  multiplicand  and  multiplier  {aX  p)\  the 
second  coefficient  is  formed  out  of  the  first  two  coefficients 
of  the  multiplier  and  multiplicand,  combined  two  by  two 
crosswise  (a  X  q  and  b  X  p);  the  third  coefficient  is  formed 
out  of  the  first  three  coefficients  of  the  multiplicand  and 
of  the  multiplier,  combined  two  by  two  crosswise  (a  X  r, 
b  X  q,  c  X  p);  and  so  on,  the  number  of  factors  of  the 
multiplicand  and  of  the  multiplier  increasing  by  one  at 
each  step  till  the  last  coefficient  of  the  multiplier  has  been 
reached. 

Then,  if  there  are  more  coefficients  in  the  multiplicand 
than  in  the  multiplier,  all  the  coefficients  of  the  multiplier 
being  retained,  the  initial  coefficients  of  the  multiplicand 
are  dropped  one  by  one,  and  a  new  one  taken  on  at  the  end, 
till  the  last  coefficient  of  the  multiplicand  has  been  reached. 


SYNTHETIC  DIVISION. 


81 


Then  one  initial  coefficient  is  dropped  from  both  multipli- 
cand and  multiplier  till  none  are  left.  In  every  case,  the 
partial  products  are  formed  out  of  the  coefficients  employed 
by  cross-multiplication. 

When  there  are  more  coefficients  in  the  multiplier  than 
in  the  multiplicand,  proceed  as  above  till  you  reach  the  last 
coefficient  of  the  multiplicand,  then,  retaining  all  the  coef- 
ficients of  the  multiplicand,  drop  the  initial  coefficients  of 
the  multiplier,  one  by  one,  and  take  in  one  at  the  end,  till 
you  reach  the  last,  and  then  drop  one  initial  coefficient 
from  both  multiplier  and  multiplicand  till  none  are  left. 
The  partial  products  are  formed  as  before  by  cross-multipli- 
cation. 

e.g.  2. 

px^  -\-  qx  -\-  r 

ax*  +  ^^^  +  '^'^^  -{-  dx  -{-  e. 


apx^  +  ':  ^5' 

x^  -\-  ar 

x'^hr 
H-  cq 

x^  +  cr 
^dq 

x^  +  dr 
-\req 

x-\-  er 

+    ^P 

\^M 

+  ci? 

+  dp 

+  ep 

Ax'  +  Bx'  +  Cx'  +  Bx^  +  Bx^  -\-  Fx  +  G. 

Note  that,  in  each  of  the  examples  just  worked  out,  the 
partial  products  cut  off  by  the  dotted  line  are  the  only  ones 
that  contain  the  first  coefficient  of  the  multiplicand  as  a 
factor,  and  that  these  partial  products  contain  this  factor 
combined  with  each  of  the  coefficients  of  the  multiplier  in 
turn. 

70.  The  Coefficients  of  the  duotient. — Hence  if  the 
product  be  taken  as  a  dividend  and  the  multiplicand  as  the 
divisor,  the  coefficients  of  the  quotient  may  be  found  by 
the  following  process.    . 


82 


DIVISION. 


1°.  In  Example  1 : 

ap  =  A,  .\  a=  A  -^ p. 

bp  =  B  —  aq,  ,',  b  =  (B  —  aq)  -^ p. 

cp  =  C  —  (ar  -\-  bq),  .'.  c  =  [C  —  (ar  -\-  bq)]  -^ p. 

Now  since  A  and  p  are  known  at  starting,  a  can  be 
found ;  then  B,  p,  a,  and  q  being  known,  b  can  be  found ; 
and  finally,  C,  p^  a,  b,  r,  and  q  being  known,  c  can  be 
found. 

2°.  In  Example  2: 


ap  —  A, 

bp  =  B  —  aq, 

cp  =  G  —  (ar-\-  bq), 

dp=  D  —  {br  -\-  cq), 

ep  =  E  —  {cr  -\-  dq), 


.  a=  A  -7-p. 

.  b  =  (B  —  aq)  -^  p. 

.c  =  [C-{ar  +  bq)]^p. 

,  d=[D-  (br-i-cq)]  -^p. 

.  e  =  [B  —  {cr  -^  cq)]  -^  p. 


In  this  case,  a,  b,  c,  d,  and  e  can  be  found  in  the  same 
manner  as  in  the  first. 

Observe  that  the  first  coefficient  of  the  quotient  is  ob- 
tained by  dividing  the  first  coefficient  of  the  dividend  by 
the  first  coefficient  of  the  divisor,  and  that  the  remaining 
coefficients  of  the  quotient  are  obtained  by  subtracting  cer- 
tain partial  products  from  the  coefficients  of  the  dividend 
which  follow  the  first,  and  then  dividing  the  remainders  by 
the  first  coefficient  of  the  divisor. 

Observe  also  that  the  partial  products  to  be  subtracted 
from  the  coefficients  of  the  dividend  are  those  above  the 
dotted  line  in  the  two  examples  worked  out,  and  that  they 
are  obtained  by  a  cross- multiplication  in  the  way  already 
described.  In  this  process  the  coefficients  of  the  quotient 
(multiplicand),  are  used  as  found,  and  only  those  coeffi- 
cients of  the  divisor  (multiplier)  which  follow  the  first  are 
employed. 


SYNTHETIC  DIVISION.  83 

If  the  signs  of  all  the  terms  of  the  divisor  which  follow 
the  first  are  reversed,  the  signs  of  the  partial  products  to  be 
subtracted  would  be  reversed,  and  the  partial  products 
would  become  additive. 

This  process  of  finding  the  coefficients  of  the  quotient 
from  those  of  the  dividend  and  divisor  is  known  as  syn- 
thetic division,  because  we  build  up  the  coefficient  of  the 
dividend  by  getting  the  partial  products  which  enter  into 
their  composition,  and  through  this  synthesis  we  obtain  the 
coefficients  of  the  quotient. 

The  following  example  will  serve  to  show  how  this  pro- 
cess may  be  carried  out  systematically. 

Divide  6a:^o  -  x^  -  V2x?  -  ^Sx^  +  18:^  -  16a;4  +  Uj^ 
+  Ux  +  4  by  2^6  -  3a;4  -  4^2  _.  2. 

First,  write  down  the  coefficients  of  the  divisor  with 
the  signs  of  all  the  terms  after  the  first  changed,  the  coef- 
ficients of  the  missing  terms  being  represented  by  zeros. 
Under  this  write  the  coefficients  of  the  completed  dividend, 
so  arranged  that  each  coefficient  may  fall  under  the  coef- 
ficient of  the  term  of  the  same  degree  in  the  divisor,  and 
as  a  matter  of  convenience  draw  a  vertical  line  after  the  first 
coefficient  of  the  divisor.  Then  obtain  the  coefficients  of 
the  quotient  by  gradually  filling  in  the  partial  products  to 
be  added  to  the  coefficients  of  the  dividend.  The  coeffi- 
cients of  the  quotient  are  written  in  the  bottom  line  to  the 
left  of  the  vertical  line,  thus : 

2 

6  +  0-  1  -  12-^ 

0  +  9+    0  +  12 

0+    0+    0 

0  +  12 

0 


3  +  0  +  4 


+ 

0  + 

3  + 

0  +  4  + 

0  +  2 

+  18- 

16  + 

24  +  0  +  12  +  4 

+ 

0  + 

6  + 

0  +  8- 

12-4 

+ 

0  + 

0  + 

0  +  0  + 

0 

+ 

0  + 

16- 

24-8 

— 

18  + 
0- 

0  + 
6 

0 

+ 

0  + 

0  + 

0  +  0  + 

0  +  0 

84  DIVISION. 

The  coefficients  in  the  last  line  are  obtained  as  follows : 

1°.  Divide  6  by  2  and  write  the  quotient  in  the  bottom 
line  under  the  first  coefficient  of  the  dividend. 

2°.  Multiply  3  by  0  (the  second  coefficient  of  the  divi- 
sor) for  the  first  partial  product,  write  the  result  under  the 
second  coefficient  of  the  dividend,  add,  divide  by  2,  and 
place  the  quotient  underneath  in  the  bottom  line. 

3°.  Form  the  next  set  of  partial  products  by  using  the 
two  coefficients  of  the  quotient  already  obtained  and  the 
two  of  the  divisor  immediately  after  the  vertical  line,  and 
multiplying  crosswise,  thus :  3x3=9  and  0x0  =  0. 
Write  these  under  the  third  coefficient  of  the  dividend, 
add,  divide  the  sum  by  2,  and  write  the  quotient  beneath  in 
the  bottom  line. 

4°.  Form  the  next  set  of  partial  products  by  using  the 
three  coefficients  of  the  quotient  already  obtained  and  the 
three  of  the  divisor  immediately  following  the  vertical  line, 
and  multiplying  crosswise,  thus:  3x0  =  0,  0x3  =  0, 
and  4  X  0  =  0.  Write  these  under  the  fourth  coefficient 
of  the  dividend,  add,  divide  the  sum  by  2,  and  write  the 
quotient  beneath. 

5°.  Form  the  next  set  of  partial  products  by  using  the 
four  coefficients  of  the  quotient  already  obtained  and  the 
four  of  the  divisor  immediately  after  the  vertical  line, 
and  multiplying  crosswise,  thus:  3  X  4  =  12,  0  X  0  =  0, 
4  X  3  =  12,  and  —6x0  =  0.  Write  these  under  the 
fifth  coefficient  of  the  dividend,  add,  divide  the  sum  by  2, 
and  write  the  result  underneath. 

We  have  now  reached  the  vertical  line  and  have  obtained 
the  coefficients  of  the  integral  part  of  of  the  quotient.  The 
remaining  part  of  the  work  is  merely  to  ascertain  whether 
or  not  there  is  a  remainder,  and  in  case  there  be  a  remain- 
der, to  obtain  its  coefficients. 

If,  on  filling  in  the  remaining  partial  products  and  add- 
ing, we  find  the  sum  to  be  zero  in  each  case,  there  is  no  re- 


SYNTHETIC  DIVISION.  85 

mainder.  If,  however,  on  filling  in  and  adding,  we  find 
the  sums  are  not  all  zeros,  there  is  a  remainder,  and  the 
sums  obtained  are  the  coefficients  of  the  corresponding 
terms  of  the  remainder.  For  the  addition  of  these  partial 
products  will  subtract  from  the  portion  of  the  dividend 
which  comes  after  the  vertical  line  the  corresponding 
portion  of  the  product  of  the  divisor  and  the  quotient 
obtained.  Hence,  if  the  result  is  zero,  there  is  no  differ- 
ence between  the  dividend  and  the  product  of  the  divisor 
and  the  quotient  obtained;  and  if  the  result  obtained  is  not 
zero,  it  must  be  the  difference  between  the  dividend  and 
the  product  of  the  divisor  and  the  quotient  obtained. 

6°.  To  obtain  the  first  set  of  partial  products  after  the 
vertical  line,  use  the  five  coefficients  of  the  quotient  already- 
obtained  and  the  five  of  those  of  the  divisor  immediately 
after  the  vertical  line,  multiplying  crosswise,  thus: 
3x0=0,  0  X  4  =  0,  4  X  0  =  0,  -  6  X  3  =  -  18,  and 
-2x0  =  0. 

7°.  To  obtain  the  next  set,  use  the  five  coefficients  of 
the  quotient  and  the  five  of  the  divisor  which  follow  the 
first  after  the  vertical  line,  thus :  3x2  =  6,  0x0  =  0, 
4  X  4  =  16,  -6x0  =  0,  -2  X  3  =  -  6. 

8°.  To  obtain  the  next  set,  omit  the  initial  coefficient 
from  each  set  used  last,  and  multiply  crosswise,  thus: 
0X2  =  0,  4X0  =  0,  -6X4=- 24,  -2X0  =  0. 

9°.  To  obtain  the  next  set,  omit  the  initial  coefficient 
from  each  set  used  last  time.  Thus:  4  X  2=8,  — 6X  0=0, 
-  2  X  4  =  -  8. 

10°.  To  obtain  the  next  set,  omit  again  the  initial  co- 
efficient, and  use  the  remainder.  Thus:  — 6x2=  —  12, 
-2x0  =  0. 

11°.  To  obtain  the  last,  omit  again  the  initial  coeffi- 
cient, and  use  the  one  remaining  in  each  set.  Thus: 
_  2  X  2  =  -  4. 


86 


DIVISION. 


The  degree  of  the  first  term  of  the  quotient  will  be  the 
difference  between  the  degrees  of  the  first  terms  of  the  divi- 
dend and  of  the  divisor,  or  4  in  this  example.  Hence  the 
quotient  is  ^x'^  +  ^^^  —  ^x  —  %. 

With  a  little  practice  the  coefficients  of  the  quotient  can 
be  obtained  with  great  ease  and  rapidity  by  this  method. 

As  a  second  example  let  it  be  required  to  divide 
i^s  _j_  ^4  _|_  3^3  _  22;2  -f  3  by  x'^  -  x^ -{- 1. 


1 

+0+1+0-1 

1+0+0+0+1 

+3 -2+0+3 

+0+1+0-1 

+0-1+0-1 

+0+0+0 

+0+0+0 

+  0  +  1 

+  0  +  1 

+  0 

+  0 

1+0+1+0+1 

+3-2+0+2 

Quotient. 

a:*  +  a;2  +  1 


Remainder. 

3a;3  _  2a:  +  2 


The  above  method  of  synthetic  division  is  applicable  to 
all  cases  of  integral  algebraic  expressions  which  contain  only 
one  letter. 

EXERCISE  XXXI. 


II. 


Exercise  XXVIII,  Examples  1-9,  15,  16,  19,  20,22-33. 


CHAPTEB  VIII. 

INVOLUTION  OP  INTEGRAL  ALGEBRAIC 
EXPRESSIONS. 

71.  Definition  of  Involution. — Involution  is  a  case  of 
multiplication  in  which  the  factors  are  all  alike.  The 
product  obtained  by  using  the  same  factor  a  number  of 
times  is  called  a  po^ver  of  the  factor.  When  the  factor  is 
used  twice  the  product  is  called  the  second  power,  or 
squa7'e  ;  when  three  times,  the  third  power,  or  cuhe  ;  when 
four  times,  the  fourth  power;  when  five  times,  the  fifth 
power;  etc. 

Involution  may  be  defined  as  the  operation  of  finding 
powers  of  numbers,  or  quantities. 

The  operation  is  indicated  by  placing  the  quantity 
within  a  parenthesis  with  an  exponent  after  it. 

Thus,  (Sa^Z*^)^  indicates  that  Zo?})^  is  to  be  cubed,  or 
raised  to  the  third  power. 

72.  Involution  of  Monomials. — Since  a  product  con- 
tains every  one  of  its  factors  as  many  times  as  each  of  these 
factors  is  contained  in  the  several  factors  counted  together, 
a  monomial  is  raised  to  a  given  power  by  raising  its  nu- 
meral coefficient  to  that  power  and  multiplying  the  exponent 
of  each  letter  by  the  exponent  of  the  given  power.     Thus : 

When  the  quantity  which  is  to  be  raised  to  any  power 
is  positive,  it  must  be  borne  in  mind  that  every  power  of  it 

87 


88  ■  INVOLUTION. 

will  be  positive,  and  that,  if  the  quantity  to  be  operated 
upon  is  negative,  every  even  power  of  it  will  be  positive 
and  every  odd  power  negative.  The  raising  of  an  expres- 
sion to  a  power  is  called  expanding  the  expression. 

EXERCISE  XXXII. 

Expand : 
1.     (ah^f,  2.     {^y^f.  3.     {;^x'^yzy. 

4.   (-  n(^dx^y.       5.   (-  ^xhff'       6.   (-  2zy)5. 

Write  down  the  square  of  each  of  the  following  expres- 
sions : 

7.     ^a%.  8.       ac^.  9.     6a%^. 

10.     -  ^a^a?.  11.     -  'la''h'x\  12.     -  'i/U'^x^, 

Write  down  the  cube  of  each  of  the  following  expres- 
sions : 

13.  M^h\         14.   -3 A.         15.   -aWx.         16.   -^/^x\ 

73.  Squaring  of  Binomials. — Any  polynomial  may  be 
squared  by  multiplying  it  by  itself;  but  it  is  easy  to  learn 
to  square  any  polynomial  at  sight. 

e.g.  (a  +  hf  =  {a^h).{a^l))  =  «2  +  <^ah  +  l^. 

a-by=(a-b).  (a  -  b)  =  a"  -  2ab  +  b^. 

X  +  3)2  ={x  +  Z).  {x  +  3)  =  a:2  +  6:?;  +  9. 

X  -  3)2  =  (2:  -  3) .  {x  -  3)  =  x^  -  6a;  +  9. 

a  +  bf  =:  {-  a-^b) .  {-  a  -\-b)  =  a'  -  %ab  +  b^, 
a-bf={-a-b).{-a-b)  =  a^  +  ^ab  +  bK 

Note  that  in  every  case  the  square  of  a  binomial  is  a 
trinomial,  and  that  two  of  the  three  terms  of  this  trinomial 
are  the  squares  of  the  two  terms  of  the  binomial  which  we 
are  squaring,  and  that  the  third  term  is  twice  the  product 


INVOLUTION.  89 

of  the  two  terms  of  the  binomial,  regard  being  had  to  the 
signs  of  the  terms.  Hence  the  following  rule  for  squaring 
a  binomial  at  sight: 

Square  each  term  of  the  bifio^nial  and  take  tioice  the 
product  of  the  tivo  terms,  and  write  the  three  terms  thus 
obtained  as  a  polynomial,  each  with  its  own  sign. 

It  is  customary  to  write  the  double  product  as  the  mid- 
dle term  in  the  result,  but  this  is  not  necessary. 

EXERCISE  XXXill. 

Write  down  the  square  of  each  of  the  following  expres- 
sions : 

I. 


1. 

«  +  35. 

2. 

a-U. 

3. 

X  —  by. 

4. 

%x  +  3y. 

6. 

Zx-y. 

6. 

3x  +  by. 

T. 

^x  -  2y. 

8. 

hab  —  c. 

9. 

pq-r. 

LO. 

X  —  abc. 

11. 

II. 
ax  -\-  %by. 

12. 

x'^  -1. 

13. 

-  4  +  a;. 

14. 

X  +  2/3«. 

15. 

X  -  2/6i. 

3« 
16.     X — .  17.     —  X  —  a.  18.     —  4  —  X. 

74.  Squaring  of  Polynomials. — 

Ex.  (a-]-b-\-cY^{a-^b  +  c)(a  +  *  +  c) 

=  ^2  +  ^,2  _^  c2  +  2«J  +  %ac  +  2^>c. 
{a-b^  cY={a-b  +  c){a  -  b  +  c) 

=  «2  ^^2_^^_  2ab  +  2ac  -  2bc. 
{a  —  b  —  c)'^  =  (a  —  b  —  c)(a  —  b  —  c) 

.     =:  «2  _|_  j2  _|_  ^2  _  2(ih  -  2ac  +  2bc. 
{—  a  —  b  —  cy=  (—a  —  b  —  c)(—a  —  b  —  c) 

=  ^2  _|_  j2  _|_  ^2  _|_  2ab-{-  2ac  +  2bc. 


90  INVOLUTION. 

Note  that  in  each  of  these  cases  the  square  consists  of 
the  square  of  each  term  of  the  polynomial  and,  in  addition, 
twice  the  product  of  the  terms  of  the  polynomial  taken  two 
by  two  in  every  possible  way,  regard  being  had  to  the  signs 
of  the  terms. 

The  surest  way  to  get  every  possible  combination  of  the 
terms  two  by  two  is  to  combine  each  term  of  the  poly- 
nomial with  each  term  which  follows  it. 

The  law  stated  above  holds  whatever  be  the  number  of 
the  terms  in  the  polynomial  to  be  squared.  Hence  we  have 
the  following  rule  for  squaring  a  polynomial : 

Square  each  term  of  the  polynomial,  and  take  twice  the 
swn  of  the  products  of  each  term  and  the  terms  which  follow 
it,  and  ivrite  the  terms  thus  obtained  as  a  polynomial,  each 
with  its  own  sign. 

EXERCISE  XXXIV. 

Form  the  squares  of: 

1.  1  -f  2^  +  3x\ 

2.  1  +  2a; -f  3a;2  +  4a;3. 

3.  1  +  2^;  +  dx^  -h  4a:3  _j_  5^5^ 
4i.    a  —  b  -{-  c  —  d. 

6.    da-i-2b  -  c-i-d. 

75.  Cubing  of  Binomials.— 

^x.(a-^bY=(a-]-b){a-\-b)(a  +  b) 

=  a^-}-  da^  +  3« J2  _|.  j3^ 

(a-bY=(a-b)(a-b)(a-  b) 

=  a^  -  da^  -h  dab^  -  b\ 

i^^  a  +  by^  (-  a-\-b)(-  a-\-  b)(-  a-i-b) 

=  -a^-i-'Sa^-'Sab^-\-b\ 
(-a-bY=(-a-b)(-a-b){~a-b) 
=  ^a^-  'da%  -  3ab^  -  b'\ 


INVOLUTION.  91 

Note  that  in  each  case  the  cube  of  a  binomial  is  a  quad- 
rinomial,  and  that  two  of  its  four  terms  aie  cubes  of  the 
two  terms  of  the  binomial,  and  each  of  the  other  two  terms 
is  three  times  the  product  of  one  of  the  terms  of  the  bino- 
mial and  the  square  of  the  other.  Hence  we  have  the  fol- 
lowing rule  for  cubing  a  binomial  : 

Cuhe  the  first  term,  take  three  times  the  product  of  the 
square  of  the  first  term  a7id  the  seco7id  term,  also  three 
times  the  product  of  the  first  term  and  the  square  of  the 
second,  and  the  cube  of  the  second  term,  and  write  the  terms 
obtained  as  a  polynomial,  each  with  its  oivn  sign. 

e.g.  (dx  -  2«2)3  ,^  {3xY-3(3xY  .  2a^-^  3(3x)(2a^y-(2a^y 

=  27a;3  -  54«V2  -|-  'SQa'x  -  Sa\ 

EXERCISE  XXXV. 

Write  down  the  cube  of  each  of  the  following  expres- 
sions : 

I. 


1.     X  +  a. 

2.     X  —  a. 

Z.     X  —  'Zy. 

4.     2x-\-2j. 

5.     3x  -  5y. 
II. 

6.     ab'  -\-  c. 

7.     2al)  —  3c. 

8.     5a  —  be. 

9.     x^  +  4?/l 

10.     4:X^  —  ^y^- 

EXERCISE  XXXVI. 

1.     Divide  9x^  - 

I. 

.Qx^-6x'  +  x'-x 

+  2  by  x^~3x-\~2. 

2.  Divide  1/43^^  +  l/72xy^  +  1/Uy^  by  l/2x  +  1/3?/. 

3.  Find  two  numbers  whose  difference  is  5,  and  such 
that  the  square  of  the  smaller  plus  9  will  equal  the  square 
of  the  laro^er  minus  56. 


92  INVOLUTION. 

4.  Find  two  numbers  which  shall  differ  by  3,  and 
such  that  the  square  of  the  smaller  plus  15  shall  equal  the 
square  of  the  larger  minus  24. 

5.  Find  two  numbers  that  shall  differ  by  2,  and  such 
that  the  cube  of  the  smaller  increased  by  six  times  its  square 
shall  be  44  less  than  the  cube  of  the  larger. 

6.  A  farmer  bought  some  cattle  at  30  dollars  a  head. 
Had  he  bought  three  more  for  the  same  money,  they  would 
have  cost  him  2  dollars  less  a  head.  How  many  did  he 
buy  ? 


CHAPTER  IX. 

EVOLUTION  OP  INTEGRAL  ALGEBRAIC 
EXPRESSIONS. 

76.  Definition  of  Evolution. — Evolution  is  the  inverse 
of  involution.  In  involution  we  have  given  the  factor  and 
the  number  of  times  it  is  employed,  and  are  required  to  find 
the  product,  or  the  power,  of  the  factor.  In  evolution  we 
have  given  the  power,  or  product,  and  the  number  of  times 
a  factor  must  be  employed  to  produce  it,  and  are  required 
to  find  the  factor. 

The  factor  whose  involution  will  produce  a  power  or 
number  is  called  the  7'oot  of  the  number,  and  the  number 
of  times  the  factor  is  to  be  employed  is  called  the  mdex  of 
the  root.  The  operation  of  finding  the  required  factor  is 
called  extracting  the  root  of  the  number. 

The  operation  of  evolution  is  indicated  by  the  radical 
sign,  V  ,  with  a  bar  extending  over  the  expression  whose 
root  is  to  be  extracted,  unless  that  expression  be  a  numeral 
or  single  literal  factor.  The  index  of  the  root  is  written  in 
front  of  the  radical  at  the  top.  Thus :  iV,  V^^.  When 
the  index  is  2  it  is  ordinarily  omitted.  A  parenthesis  may 
be  used  in  any  case  instead  of  the  bar. 

77.  Inverse  of  Involution.  —  Involution  is  not  com- 
mutative, that  is,  2^  does  not  equal  5^.  In  subtraction,  the 
inverse  of  addition,  there  are  two  questions  that  may  be 
asked.  For  example,  we  may  ask  what  number  must  be 
added  to  5  to  make  9,  or  to  "what  number  must  5  be  added 


94:  EVOLUTION. 

to  make  9 ;  but  as  addition  is  commutative,  there  is  only 
one  inverse  operation.  Each  of  the  above  questions  is  an- 
swered by  subtraction. 

Also  in  division,  the  inverse  of  multiplication,  two 
questions  may  be  asked.  For  example,  we  may  ask  how 
many  times  is  4  contained  in  20,  or  what  number  is  con- 
tained 4  times  in  20.  This  is  equivalent  to  asking  *'20  is 
how  many  times  4,  or  20  is  4  times  what  number.  ^^  But 
since  multiplication  is  commutative,  there  is  only  one  in- 
verse operation.  Each  of  the  above  questions  is  answered 
by  division. 

In  evolution,  the  inverse  of  involution,  two  questions 
may  likewise  be  asked.  For  example,  we  may  ask  what  is 
the  fifth  root  of  32,  or  what  root  of  32  is  2.  As  involution 
is  not  commutative,  these  questions  cannot  be  answered  by 
one  and  the  same  operation.  The  former  is  answered  by 
evolution,  and  the  latter  by  logarithms.  The  former  is  the 
only  inverse  operation  that  we  shall  consider  here. 

78.  Corresponding  Direct  and  Inverse  Operations  do 
not  always  Cancel  each  Other.  —  Corresponding  inverse 
and  direct  operations  usually  cancel  each  other.  Thus  the 
addition  and  subtraction  of  the  same  number  cancel  each 
other,  the  multiplication  and  division  by  the  same  number 
cancel  each  other,  also  the  extraction  of  a  root  and  raising 
to  the  corresponding  power  cancel  each  other.     Thus : 


It  must,  however,  be  borne  in  mind  that  roots  are  more 
than  one- valued,  and  hence  the  statement  with  reference  to 
the  inverse  operations  of  extracting  roots  and  raising  to 
powers  need  restriction.  It  is  true,  "necessarily  and  uni- 
versally, that  [l^aY  =  a,  but  not  that  \/a''  =  a.  For 
instance,  Va^  =  "^a.  Wliile  the  statement  that  the  extrac- 
tion of  a  root  is  cancelled  by  raising  the  result  to  the  cor- 


EVOLUTION.  95 

responding  power  is  true  necessarily  and  universally,  the 
inverse  statement  that  the  raising  an  expression  to  a  power 
is  cancelled  by  the  extraction  of  the  corresponding  root  of 
the  result  is  not  necessarily  true. 

79.  Extraction  of  Roots  of  Monomials. — Since  evolution 
is  the  inverse  of  involution,  we  extract  the  root  of  an  ex- 
pression by  doing  just  the  opposite  to  what  we  do  in  finding 
a  power. 

Thus,  we  find  the  power  of  a  monomial  by  raising  its 
numeral  factor  to  the  power  indicated  by  the  exponent,  and 
multiply  the  exponent  of  each  literal  factor  by  the  exponent 
of  the  power. 

e.g.  (4a;V)3  :=  UxhK 

Hence  we  extract  the  root  of  a  monomial  by  extracting 
the  indicated  root  of  the  numeral  factor  and  dividing  the 
exponent  of  each  letter  by  the  index  of  the  root. 


e.g.  V^^x^z^  —  4A^ 

N.B.  — Since  (=^«)^  =  6^2,  .'.   Vd' =  "^a. 

That  is,  the  square  root  of  a  positive  quantity  is  either 
-f-  or  — ,  and  the  square  root  of  a  negative  quantity  is  im- 
possible, or  imaginary.    The  same  is  true  of  any  even  root. 

The  odd  root  of  a  positive  quantity  is  -{-,  and  of  a 
negative  quantity  — . 

EXERCISE  XXXVII. 

I. 
Find  the  indicated  roots  of  the  following  monomials : 


1.      Va^h^c'\ 


3.      Vlla^1)'c\  4.      f-  Uda'^'^ 


5,      Vx''yH\  6.      V'-x'Y^ 


96  EVOLUTION. 

80.  Extraction  of  the  Square  Root  of  Polynomials. — 

To  obtain  a  rule  for  extracting  the  root  of  a  polynomial,  let 
us  examine  the  square  of  a  polynomial. 

e.g.  (rt  +  J  +  c  +  ^0' 
=  a^  +  ^,2  ^  ^2  +  6?2  +  ^ab  +  %ac-^'^ad-\^Uc^^hd-]-^cd 
=  a'  +  2ah  +  *^  +  ^ac  +  2bc  +  c^  +  2ad -\- 2M -\- 2cd -}- d'' 
=  a^-\-(2a-\-b)b-]-{2a-{-2b-\-c)c+(2a-{-2b-\-2c-\-d)d 
=  a^  -{-  {2a  -]-  b)b  +  [2(a  +  b)  -\-  c]c-{-ma-\-b  -}-  c)  -{-  d]d. 

From  the  last  of  the  above  equations  we  may  derive  the 
following  rule  for  writing  at  sight  the  square  of  any  poly- 
nomial : 

Write  the  square  of  the  first  term,  then  the  product  of 
twice  the  first  term  pl^is  the  second  multiplied  by  the  second, 
then  the  product  of  twice  the  first  tivo  terms  plus  the  third 
multiplied  by  the  third,  then  the  product  of  twice  the  first 
three  terms  plus  the  fourth  multiplied  by  the  fourth,  etc. 

If  now  we  take  the  second  member  of  the  second  equa- 
tion and  compare  it  with  the  second  member  of  the  last, 
we  may  readily  obtain  a  rule  for  extracting  the  root  of  a 
polynomial. 


a^-{-2ab-^b''-{-2ac-Jf-2bc-\-c''+2ad-\-2bd-{-2cd-{-d'^  I  a+b-]-c-^d 


2a-{-b 


2ab-{-b^ 
2ab  4-  6' 


2a  4-  2&  +  c 


2ac  +  3&C  4-  c« 
2ac  4-  26c  4-  c^ 


2a^2b-{-2c-\-d 


2ad  +  2bd  +  2cd  +  d^ 
2cd  4-  2bd  4-  2cd  +  d^ 


First  arrange  the  terms  of  the  p)olynomial  according  to 
the  powers  of  some  letter  ;  theii  tahe  the  square  root  of  the 
first  term,  place  it  in  the  root  or  quotient,  square,  subtract, 
and  bring  dotvn  one  or  more  terms;  then  double  the  root 


EVOLUTION.  97 

already  found  and  place  the  resiilt  in  the  divisor,  find  how 
many  times  this  is  contained  in  the  first  term  of  the  re- 
7nainder,  place  the  result  in  both  the  root  atid  in  the  divi- 
sor, multiply,  subtract^  and  briny  dotvn;  then  double  the 
root  already  found  and  proceed  as  before  ;  and  so  on  to  the 
end. 

EXERCISE  XXXVIII. 

Extract  the  square  roots  of: 

I. 

1.  a^  +  4:a^  +  2a^  -  4a  +  1. 

2.  x^  -  2x^y  +  3xY  -  '^xy^  -\- t/,       . 

3.  4ft«  -   rZa^X  -\-  5ff4^2  _|_  6^3^3  _^  ^2.^4_ 

4.  9x^  -  12.cy  +  IQxY  -  24:xy  +  4?/6  +  lQxy\ 

5.  4^8  +  166'8  +  IQa'c^  -  32«V. 

6.  4:X^-}-9  -  SOx  -  20ic3  +  37:^2. 

7.  162;^  -  Uabx^  +  16^»2a;2  +  la^b'^  -  8ab^  +  4:b\ 

II. 

8.  x^  +  25x^  +  10^-4  -  4a;5  -  20a^  +  16  -  24:r. 

9.  a;^  +  SxY  —  ^^y  —  ^xy^  +  %xh/  —  lOxY  +  y^- 

10.  4  -  12a  -  lla^  +  5^2  _  4«5  4-  4a«  +  14a^. 

11.  25a;«  -  'dWy'^  +  34a;y  _  'dOx^y^y^-^xy^^lQxY- 

12.  ^c"^  —  a;^?/  —  7/4a:y  +  a:^^  +  ?/*. 

13.  x^  -  4:a^y  +  6xY  -  Qxy^  +  5/  -  ?^  +  ^. 

81.  Squaring  Numbers  as  Polynomials. — Every  number 
composed  of  two  or  more  digit's  may  be  written  as  a  poly- 
nomial.    Thus:  25  =  20  +  5,  234  =  200  +  30  +  4,  etc. 


98  EVOLUTION. 

Hence  (234)2  =  (200  +  30  +  4)'^ 

=  (200)2+(2.2004-30)x30+(2.230+4)4. 
40000  +  12900  +  185G  =  5475G. 

EXERCISE  XXXIX. 

In  a  similar  way  find  the  squares  of  the  following 
numbers : 

I. 

1.     327.  2.     3789.  3.     845. 

II. 

4.    5006.  5.     19683.  6.     5083. 

Observe  that  the  square  of  a  number  contains  either 
twice  as  many  or  one  less  than  twice  as  many  places  as  the 
number  itself. 

Ex.  .234    =  .2  +  .03  +  .004. 

(.234)2  =  (.2 +  .03 +  -004)2 

=  (.2)2+(2x.2+.03).03+(2x.23+.004).004 

=  .054756. 

In  a  similar  way  find  the  square  of : 

I. 
7.     .0304.  8.     .0028. 

Observe  that  when  a  number  is  a  decimal,  its  square  is 
a  decimal  and  contains  twice  as  many  places  as  the  num- 
ber. 

Ex.   23.4  =  20  +  3  +  .4. 

(23.4)2=3(20+3+. 4)2=(20)2+(2x20+3)3+(2x23+.4).4 
^  547.56. 


EVOLUTION. 
In  a  similar  way  find  the  squares  of : 


I. 

9. 

69.4. 

10. 

II. 

43.21. 

11. 

37.89. 

12. 

8.008. 

Observe  that  when  the  number  is  composed  of  an  integer 
and  a  decimal,  its  square  is  composed  of  an  integer  and  a 
decimal,  and  that  the  number  of  places  in  the  integral  part 
of  the  square  is  either  twice  as  great  or  one  less  than  twice 
as  great  as  that  in  the  integral  part  of  the  number,  and  in 
the  decimal  part  of  the  square  twice  as  great  as  in  the  deci- 
mal part  of  the  number. 

82.  Extracting  the  Square  Root  of  Numbers. — Observe, 
in  all  the  cases  of  the  last  section,  that  if  we  begin  at  the 
decimal,  point  and  divide  the  square  into  periods  of  two 
places  each,  the  square  root  of  the  largest  square  in  the  left- 
hand  period  will  be  the  left-hand  figure  of  the  number 
squared,  and  the  number  of  this  left-hand  period,  counting 
from  the  decimal  point,  will  be  the  order,  or  place,  of  the 
figure  in  the  root,  or  in  the  number  squared. 

Hence  the  first  step  in  finding  the  root  of  a  number  is 
to  divide  the  number  into  periods  of  two  figures  each,  be- 
ginning at  the  decimal  point. 

The  periods  thus  obtained  correspond  to  the  terms  of  a 
polynomial  whose  square  root  is  to  be  found,  and  the  pro- 
cess of  finding  the  square  root  of  a  number  is  precisely 
analogous  to  that  of  finding  the  square  root  of  a  poly- 
nomial. 


e.g.    |/387420489. 


100 


EVOLUTION. 


3  -  87  -  42  -  04  -  89il0000  +  9000  -f  600  +  80  +  3 
1  00  00  00  oo' 


20000  +  9000 
29000 

2 
2 

87 
61 

42 

00 

04 
00 

89 
00 

38000  +  600 
38600 

26 
23 

42 
16 

04 
00 

89 
00 

39200  +  80 
39280 

3 
3 

26 
14 

04 
24 

89 
00 

39360  +  3 
39363 

11 
11 

80 
80 

89 
89 

19683. 


It  appears  from  the  above  example  that,  after  the  first 
step,  the  extraction  of  the  square  is  a  case  of  division,  in 
which  the  divisor  varies  with  each  remainder,  and  in  which 
the  exact  or  complete  divisor  is  unknown.  It  also  appears 
that  the  incomplete  or  trial  divisor  in  each  case  is  double 
the  part  of  the  root  already  found. 

Evidently  the  work  in  the  above  example  might  be 
made  more  compact  by  omitting  the  ciphers,  and  writing 
the  root  at  once  in  the  usual  form,  instead  of  in  the  form  of 
a  polynomial.     Thus : 


3- 
1 

-87  - 

-42- 

-04- 

-89 

19683 

29 

2 
2 

87 
61 

386 

1 

26 
23 

42 
16 

3928 

~  3 

26 
14 

04 
24 

39263 

1 

11 
11 

80 
80 

89 
89 

From  the  above  considerations  we  may  deduce  the  fol- 
lowing rule  for  extracting  the  square  root  of  a  number: 


EVOLUTION.  Ml 

Divide  the  number  into  periods  of  two  places  each,  be- 
ginning at  the  deci7nal  point;  find  the  largest  perfect 
square  in  the  left-hand  period,  subtract  it  from  this  period 
and  place  its  root  in  the  quotient,  and  bring  down  the  next 
period;  double  the  root  already  found  for  a  trial  divisor, 
and  seek  how  many  times  this  is  contained  in  the  remainder 
■exclusive  of  the  last  figure,  and  place  the  result  in  both  the 
divisor  and  the  quotient;  multiply,  subtract,  bring  down, 
and  proceed  as  before. 

As  the  trial  divisor  is  smaller  than  the  real  divisor,  we 
must  guard  against  taking  too  large  a  figure  for  the  quo- 
tient.    Of  course  this  figure  can  never  exceed  9. 

Should  the  trial  divisor  not  be  contained  in  the  remain- 
der after  the  last  figure  has  been  excluded,  place  a  cipher 
in  the  divisor  and  quotient,  and  bring  down  the  next  period 
and  try  again,  and  so  on  till  a  significant  figure  is  obtained. 

In  the  actual  work,  after  the  number  has  been  separated 
into  periods,  the  decimal  points  may  be  disregarded.  It 
should  be  placed  in  the  quotient,  or  root,  when  its  position 
has  been  reached,  but  farther  than  this  it  may  be  entirely 
neglected. 

When  the  number  is  not  an  exact  square,  its  root  may 
be  obtained  to  any  required  degree  of  approximation  by 
bringing  down  two  ciphers  for  each  new  period.  Of  course 
care  must  be  taken  to  place  the  decimal  point  in  the  right 
position  in  the  quotient. 

EXERCISE  XL. 

Find  the  square  roots  of : 

I. 

1.  14356521.        •  2.  25060036. 

3.  25836889.        4.  16803.9369,  1 


W^ii       ■':'  EVOLUTION. 

II. 

6.  4.54499761.  6.     .9. 

7.  6.21.  8.     .00852. 

83.  Cubing  of  Polynomials. — 

{a  +  bf  =  «3  +  3«2^  +  'dah^  +  h^ 

(«  +  Z»  +  6f   =  «3  +  ^,3  _^  c^  4_   3«2^  4.   3^2^  _|_   3^,2^  _^   3^^2 

+  3«c2  +  3^c2  +  6«Jc 

=  ««  +  (3«2  +  3«^  +  V")}}  +  [3(«  +  Z*)2  + 
3(«  +  ^)c  +  o'^c. 

By  means  of  the  above  formulas  the  cube  of  any  poly- 
nomial may  be  written  at  sight.  First,  write  the  cube  of 
the  first  term ;  then  the  product  of  three  times  the  square 
of  the  first  term  plus  three  times  the  product  of  the  first 
and  second  terms  plus  the  square  of  the  second  term  multi- 
plied by  the  second;  then  the  product  of  three  times  the 
square  of  the  first  two  terms  plus  three  times  the  product 
of  the  first  two  terms  and  the  third  plus  the  square  of  the 
third  multiplied  by  the  third;  etc. 

EXERCISE  XLI. 

Cube  the  following  polynomials  by  the  above  method : 

I.  . 

1.     a-\-\.  2.     X  -\-  2.  3.     ax  —  y^. 

4.     2m  -  1.  5.     4a  -  U.         e.     1  +  a:  +  x\ 

n. 

7.    1  -  2a;  +  ^x\  8.    a-\-U-c. 

9.    U^  -  3a  +  1.  IQ,     l-x-\-x'^  -  x\ 


EVOLUTION.  103 

84.  Extracting  the  Cube  Root  of  Polynomials. — If  we 

arrange  the  terms  of  {a-\-  b  -{-  cY  according  to  the  descend- 
ing powers  of  a  and  the  ascending  powers  of  c,  we  have 

Comparing  this  with 

a'  +  (3«2  -f-  Mb  +  b^)b  +  \^{a  -\- bf -^  Z(a -\-  b)c  +  c^c, 

we  may  readily  extract  the  cube  root  of  the  first  expression. 
Thus: 


a8_|_3^25^3^;,2_|.53_|_3«2c_^6«jc+3&2c4-3ac2-j-3&c2+c3|a+6+c 


3a»+3a6+62 


3«*6+3a*2+63 


3«--»+6a6+362+3ac+36c+c- 


3<z2c-j-6a6c+36-'c + 3«cH  36c-^-|-c» 


The  rule  for  extracting  the  cube  root  of  a  polynomial 
may  be  stated  as  follows : 

Arrange  the  terms  according  to  the  powers  of  some  letter 
or  letters  ;  extract  the  cube  root  of  the  first  term  and  place 
the  root  in  the  quotient  and  suMract  the  cuhe  from  the  poly- 
nomial, and  bring  do2vn  a  part  of  the  remainder  ;  use  three 
times  the  square  of  the  root  already  found  as  a  trial  divi- 
sor, and  seek  hoiv  many  times  this  is  contained  in  the  first 
term  of  the  remainder,  place  the  result  as  a  7ieiu  term  in  the 
quotient,  and  place  three  times  the  product  of  this  term  and 
the  root  already  found,  and  also  the  square  of  this  term,  as 
a  new  term  in  the  divisor,  multiply,  subtract,  and  bring 
dow7i;  a7id  so  on  till  there  is  no  remainder,  or  u?itil  the 
desired  degree  of  app)roximation  has  been  reached. 


104  EVOLUTION. 

EXERCISE  XLII. 

Find  the  cube  roots  of: 

I. 

1.  1  -  3a:  +  32;2  -  x^. 

2.  1  +  6^  +  122;2  +  Sx^. 

3.  ^x^  -  ^^x^y  +  54x^2  _  27^/3. 

4.  272;%3  -  21x^yH^  +  9:r^;2^  -  zK 

II. 

5.  24«2J  +  «3  _^  hlW  +  192«R 

6.  108:^5  -  144a:*  -  27a:«  +  64ar^. 

7.  1  +  3a:  +  6a:2  +  7a:3  _^  g^,4  _j_  3^^5  _|.  ^6^ 

8.  1  —  a:  to  four  terms. 

85.  Cubing  Numbers  as  Polynomials. — Any  number 
may  be  written  as  a  polynomial  and  then  cubed  by  the 
method  of  83.     Thus: 

1854  =  1000  +  800  +  50  +  4, 
and  (1854)3=  (1000  +  800  +  50  +  4)^ 

=  10003+  [3(10002 +  1000  x800)+8002]800+ 
[3(18002+1800x50)+502]50+[3(18502+1850x4)+42]4 
=  lOOpOOOOOO  +  4  832  000  000  +  499  625  000  +  41 158  864 
=  6  372  783  864. 

EXERCISE  XLIIf. 

Cube  the  following  numbers  by  the  process  of  46 : 

I. 
1.     135.  2.     223.  3.     106. 

4.    258.  6.     478.  6.     46.8. 

II. 
7.     9.36.  8.    27.55.  9.     .384. 


EVOLUTION.  105 

86.  Extracting  the  Cube  Root  of  Numbers. — Observe 
in  each  of  the  above  cases  that  the  cube  of  a  number  con- 
tains three  times  as  many  figures  as  the  number  cubed,  or 
one  or  two  less  than  three  times  as  many;  that  when  the 
number  cubed  is  an  integer,  the  cube  is  an  integer;  that 
when  the  number  cubed  is  a  decimal,  the  cube  is  a  decimal ; 
that  when  the  number  cubed  is  composed  of  an  integer 
and  a  decimal,  the  cube  is  also  composed  of  an  integer  and 
a  decimal. 

Observe  also  that  if  we  divide  the  cube  into  periods  of 
three  places  each,  beginning  at  the  decimal  point,  the 
number  of  periods  in  the  cube  will  equal  the  number  of 
figures  in  the  number  cubed ;  and  that  the  cube  root  of  the 
largest  cube  in  the  left-hand  period  will  be  the  left-hand 
figure  of  the  number  cubed. 

Hence  the  first  step  in  finding  the  cube  root  of  a  num- 
ber is  to  divide  the  number  into  periods  of  three  figures 
each,  beginning  at  the  decimal  point. 

The  periods  thus  obtained  correspond  to  the  terms  of  a 
polynomial  whose  cube  root  is  to  be  found,  and  the  process 
of  finding  the  cube  root  of  a  number  is  precisely  analogous 
to  that  of  finding  the  cube  root  of  a  polynomial. 

e.g.  Extract  the  cube  root  of  12  977  875. 

12-977-875  |  200  +  30  +5  =  235 
2003  ^  8  000  000 


3x2002  =  120000 

3x200x30=    18000 

30'^  =   900 


4  977  875 
4  167  000 


138900 
3  X  2302  ^ 158700 
3X230X5=   3450 
52=    25 

162175 


810  875 
810  875 


106  EVOLUTION. 

It  appears  from  the  above  example  that,  after  the  first 
step,  the  extraction  of  the  cube  root  is  a  case  of  division, 
in  which  the  exact  or  complete  divisor  is  unknown.  It 
also  appears  that  the  incomplete  or  trial  divisor  in  each 
case  is  three  times  the  square  of  the  part  of  the  root  already 
found. 

As  in  square  root,  the  process  may  be  made  more  com- 
pact, by  omitting  the  ciphers  in  the  root,  and  writing  it  at 
once  in  the  usual  form.  The  ciphers  may  also  be  omitted 
from  the  partial  subtrahends,  and  only  one  period  need  be 
brought  down  at  a  time.  One  cipher  must,  however,  be 
employed  for  the  next  place  in  finding  the  trial  and  com-, 
plete  divisors.  This  is  necessary  because  the  significant 
figures  in  the  additions  to  the  trial  divisor  often  overlap 
those  of  the  trial  divisor. 

As  regards  decimal  points  and  imperfect  cubes,  the 
same  remarks  apply  as  to  square  root. 

As  the  trial  divisor  in  cube  root  is  considerably  smaller 
than  the  real  divisor,  there  is  great  liability  to  make  the 
next  figure  too  large,  and  the  right  figure  often  can  be 
ascertained  only  after  two  or  three  trials. 


EXERCISE  XLIV. 

Find  the  cube  roots  of 

: 

1.     109  215  352. 

I.     . 

2.     56.623  104. 

3.     102.503  232. 

4.     820.025  856. 

6.     20  910.518  875. 

6.     2.5. 
II. 

7.     .2. 

8.      .01.                             9. 

4. 

10.     .4. 

U.     28.25.                   12. 

15f. 

EVOLUTION.  107 

I. 

13.  Divide  27«V  -  8jy  by  3^2^  -  2%. 

14.  {x  +  If  -  {x^  -I)-  x(2x  +  1)  -  2(x  +  2)(x  +1) 

+  20. 

15.  The  length  of  a  room  exceeds  its  breadth  by  3  ft. 
Were  its  length  increased  by  3  feet  and  the  breadth  dimin- 
islied  by  2  feet,  the  area  of  the  room  would  remain  the 
same.     Find  the  dimensions  of  the  room. 

II. 

16.  Divide  8a^  +  64c«  by  4^^  -  Sa^c^  +  IQcK 

17.  25a:  -  19  -  [3  -  i4:X  -  5)]  =  Sx  -  (Qx  -  5). 

18.  The  length  of  a  room  exceeds  its  breadth  by  8  ft. 
Were  each  increased  by  2  feet,  it  would  take  26|  yards 
more  of  carpeting  3/4  of  a  yard  wide  to  cover  the  floor. 
Find  the  dimensions  of  the  room. 

19.  In  a  cellar  one  fifth  of  the  wine  is  port  and  one 
third  claret.  Besides  this  it  contains  15  dozen  bottles  of 
sherry  and  30  bottles  of  spirits.  How  many  bottles  of  port 
and  of  claret  does  it  contain  ? 

20.  A  boy  bought  some  apples  at  three  a  cent  and  5/6 
as  many  at  four  a  cent.  He  sells  them  at  16  for  6  cents 
and  gains  3^  cents.     How  many  apples  did  he  buy  ? 


CHAPTER  X. 
MULTIPLICATION  AT  SIGHT. 

87.  Complete  Algebraic  Expressions. — A  complete  al- 
gebraic expression  of  the  first  degree  in  any  one  letter  is  a 
binomial,  one  of  whose  terms  contains  the  first  power  of 
the  letter  and  the  other  does  not  contain  the  letter  at  all. 
Thus.  X  -\-  5,  dx  —  a  are  complete  expressions  of  the  first 
degree  in  x. 

The  term  of  an  expression  which  does  not  contain  the 
letter  or  unknown  quantity  is  called  the  constant  or  absolute 
term. 

A  complete  algebraic  expression  of  the  second  degree  in 
any  one  letter  is  a  trinomial,  one  of  whose  terms  contains 
the  second  power  of  the  letter,  another  the  first  power  of 
the  letter,  and  the  third  does  not  contain  the  letter  at  all. 
Thus,  x^  -{-  6x  —  6,  dx^  —  4:X  -\-  a  are  complete  expressions 
of  the  second  degree  in  x. 

88.  Product  of  Two  Binomials  of  the  First  Degree. — 
The  product  of  two  binomial  expressions  of  the  first  degree 
in  any  letter  is  generally  a  trinomial  of  the  second  degree 
in  that  letter,  though  it  is  in  one  case  a  quadratic  binomial. 
The  student  should  be  able  to  write  with  facility  at  sight 
the  product  of  any  two  first-degree  binomials  in  the  same 
letter. 

Suppose  we  are  required  to  obtain  the  product  of  3x-\-4: 
and  5x—  7.  The  literal  factor  of  the  first  term  will  be  x^, 
of  the  second  term  x,  and  the  third  term  will  not  contain  x. 

108 


MULTIPLICATION  AT  SIGHT  109 

The  annexed  diagrammatic  arrangement  will  enable 
us  to  obtain  the  coefficients. 

The  coefficients  are  to  be  multiplied  together  as  indicated 
by  the  connecting  lines.  The  product 
of  the  left-hand  coefficients  will  be  ^ 
the  coefficient  of  x^,  the  sum  of  the 
two  cross-products  will  be  the  coeffi- 
cient of  X,  and  the  product  of  the 
right-hand  factors  will  be  the  absolute  ^ 
term.  Care  must  be  taken  to  use  the 
right  sign  with  each  coefficient  of  x  and  with  the  absolute 
term,  and  also  with  each  product. 

The  product  of  the  above  binomials  will  be  found  to  be 
Ibx^  -  a;  -  28. 

We  would  advise  using  the  diagrammatic  arrangement  in 
all  cases  at  first  till  the  pupil  has  acquired  facility  in  obtain- 
ing the  new  coefficients.  The  diagram  may  then  be  dis- 
carded, and  the  product  written  down  at  once,  the  work  of 
obtaining  the  result  being  entirely  mental. 

EXERCISE  XLV. 

Find  by  the  above  method  the  products  of  the  following 
pairs  of  first-degree  binomials: 


1.  2a;  —  5  and  Ix  —  4. 

3.  4  —  5x  and  7  —  ^x. 

6.  x  -\-l  and  x  -\-  9. 

7.  X  —  ^  and  x  -\-^. 
9.  X  -\-  b  and  X  —  Q. 

11.  X  -\-  ^  and  X  +  3. 

13.  X  -\-  %  and  a;  —  8. 


I. 

2. 

5rc  -f-  8  and  ^x  +  6. 

4 

6  +  8:2;  and  5  -  lOx, 

6. 

X  ~  5  and  x  —  S. 

8. 

2;  —  11  and  X  +  7. 

10. 

X  -\-  7  and  x  —  4. 

12. 

X  —  4:  and  x  —  4. 

14. 

X  —  Q  and  x  -(-  6. 

110  MULTIPLICATION  AT  SIGHT. 

II. 

15.  7x-  9  and  6x  +  12.       16.     6:c  -  3  and  12x  +  8. 

17.  'Sx  +  7  and  8^  -  25.       18.     2x  +  6  and  9x  -  30. 

19.  ax  —  b  and  4:X  —  5.         20.     'Sax  -\-  c  and  Qx  -\-  8. 

21.  ax  —  c  and  5cfa:  +  ^• 

22.  (a  +  ^)i^  +  c  and  2ax  —  b. 

23.  3  —  9:?;  and  8  +  12a:.       24.     9  +  4:?;  and  7  —  8a;. 

89.  Product  oix  -\-  a  and  x  +  J. — Observe  in  examples 
5-10  that  when  the  coefficient  of  x  in  the  factors  is  unity, 
the  coefficient  of  x^  in  the  product  will  be  unity,  that  the 
coefficient  of  x  in  the  product  will  be  the  algebraic  sum  of 
the  constant  terms  of  the  factors,  and  that  the  constant 
term  in  the  product  will  be  the  algebraic  product  of  the 
constant  terms  of  the  factors.  Also  that  the  constant  term 
of  the  product  will  be  positive  when  the  constant  terms  of 
the  factors  have  like  signs,  and  negative  when  the  constant 
terms  of  the  factors  have  unlike  signs,  and  that  the  sign  of 
the  term  in  x  in  the  product  is  that  of  the  constant  term  of 
the  factors  which  is  numerically  the  larger. 

The  cases  illustrated  by  these  six  examples  are  of  very 
common  occurrence,  and  careful  attention  should  be  given 
them. 

90.  Product  oi  X  -\-  a  and  x  +  a. — Observe  in  examples 
11  and  12  that  when  the  two  factors  are  alike,  the  result  is 
the  same  as  that  obtained  by  the  formula  for  squaring  a 
binomial. 

91.  Product  oi  X  -\-  a  and  x  —  a. — Observe  in  examples 
13  and  14  that,  when  the  corresponding  terms  of  the  two 
binomial  factors  are  alike  in  absolute  value  but  different  in 
their  connecting  sign,  the  product  is  a  hinomial,  and  that 
the  two  terms  of  the  product  are  the  squares  of  the  corrc- 


MULTIPLICATION  AT  SIGHT  111 

sponding  terms  of  the  factors,  and  that  the  sign  between  the 
terms  of  the  product  is  minus. 

This  is  the  only  case  in  which  the  product  of  two  bi- 
nomials is  a  binomial.  In  all  other  cases  it  is  a  trinomial. 
This  case  is  particularly  important,  and  is  known  as  the 
*' product  of  the  sum  and  difference  of  two  quantities,"  and 
is  usually  stated  thus: 

The  product  of  the  sum  and  difference  of  tivo  quantities 
is  equal  to  the  difference  of  their  squares. 

92.  Product  of  any  Two  Binomial  Factors  of  the  same 
Degree. — Any  two  binomial  factors  which  are  of  the  same 
degree  in  the  same  letter,  and  each  of  which  has  a  constant 
term,  may  be  multiplied  at  sight  by  the  method  of  section 
88.  The  literal  factor  in  one  term  of  the  product  will  be 
the  square  of  the  factor  in  the  given  binomials,  in  another 
term  of  the  product  it  will  be  the  same  as  in  the  given  bi- 
nomials, and  in  the  third  term  of  the  product  it  will  not 
occur  at  all.  The  coefficients  and  con- 
stant term  of  the  product  may  be  ^^\  /^^ 
found  by  the  diagrammatic  arrange- 
ment given  in  section  88. 

Ex.  Find  the  product  of  dx^  +  5 
and  4r^  -  8. 

Ans.   nx^  -  4^3  _  40. 

EXERCISE  XLVI. 

Write  at  sight  the  products  of  the  following  pairs  of 
binomials : 

I. 
1.     4:c2  -  7  and  bx^  -  3.  2.     7cc*  +  4  and  Sa;^  +  5. 

3.     bx^  +  4  and  ^x^  -  8.  4.     ^^  -  2  and  1':^  +  3. 

6.  Zz^  -  8  and  72^  +  12.         6.     9?/^  -f-  11  and  6/  -  7. 

7.  Vx^bm^  Vx-\-l.        8.     %Vx-Qm^zVx-\-^, 


13. 

7?i*  -  2  and  6m^  -  8. 

16. 

s^-7  and  s^  +  8. 

17, 

x'-7  and  x'  +  7. 

19. 

'Za^  -  4  and  2x^  +  4. 

21. 

3V^+5  VfandSV: 

112  MULTIPLICATION  AT  SIGHT. 

9.      Vx-7  and  i/^+  7.       lO.     d  Vx -{- 4:  and  3  Vx -\-  4. 

11.  :?^  +  V5  and  x  —  V5. 

12.  V//i  +  VS  and  l/m  —  V~5. 

II. 
14.     ^i^  +  12  and  3;^^  -  15. 
16.     a^  -f  *J  and  a^  —  11. 
18.     '?^^^  +  6  and  m^  —  6. 
20.     5 aV  —  3  and  5a V  +  3. 
■  6V7- 

22.  6  V^- 7  V3  and  6  t/a;  +  7|/3. 

23.  a;  +  V—  4  and  :?;  —  V—  4. 

24.  2a;2  +  3  V^^5  and  2^:2  -  3  V^^. 

93.  Products  of  Binomial  Aggregates. — Any  aggregate 
may  take  the  place  of  the  literal  factor  in  the  preceding  bi- 
nomials and  the  product  obtained  by  the  same  methods. 
Of  course  the  aggregate  must  have  the  same  exponent  or 
radical  index  in  the  two  binomial  factors. 

EXERCISE  XLVII. 

Write  at  sight  the  product  of  the  following  pairs  of 
binomials : 

I. 

1.  (a-\-  x)  -{-  4:  and  (a -\-  x)  —  7. 

2.  (m  +  ^)  —  8  and  (m  -{-x)-\-^.      ' 

3.  {x  —  h)  —  ^  and  {x  —  b)  -\-  9. 

4.  (x  —  m)  —  12  and  (x  —  m)  -\-  7. 

5.  X  —  |/(m  —  5)  and  x  +  \/{m  —  5). 

6.  X  -[-  |/(3  —  a)  and  x  —  |/(3  —  a). 


MULTIPLICATION  AT  SIGHT.  113 

7.  {x  —  A)  -\-  {x  —  a)  and  {x  —  4)  —  (:r  —  a). 

8.  V{^^  +  2^^  +  3^^  +  4a:3  +  3x2  +  2a;  +  1)  =  ? 

9.  |/(1  -  9rr2  +  33a;4-63x6_|_66^8_36^io_|_  8a;i2)^  ? 

-^/(a;  -  6)  a;  +  8  ^      .         ^, 

10.  -i^^-i  -  ^_^^- :.  5  V(.  -  6). 

11.  A  alone  can  do  a  piece  of  work  in  nine  days,  and 
B  alone  in  12  days.  How  many  days  will  it  take  them  to 
do  it  together  ? 

12.  A  cistern  could  be  filled  in  12  minutes  by  two 
pipes  which  empty  into  it,  and  it  could  be  filled  in  20 
minutes  by  one  of  these  pipes  alone.  How  many  minutes 
would  it  take  the  other  pipe  alone  to  fill  it  ? 

II.* 

13.  (a:  -  5)  +  (ic  +  6)  and  {x  -  b)  -  {x -\-  6). 

14.  {x-Yl)-{x-  5)  and  {x-\-l)^{x-  5). 

15.  V{^  +  2)  +  5  and   ^{x  +  2)  -  5.       . 

16.  Vi^  -  ^)  +  4  and  ^{x  -  7)  -  4. 

17.  X  +  V{^  ~  ^)  ^^^  ^  ~  V{^  —  5)- 

18.  |/(^  +  4)  +  \/{x  -  7)  and  ^{x  +  4)  -  ^{x  -  7). 

19.  Vi^  +  8)  +  ViP^  +  5)  and  ^{x  +  8)  -  \/{x  +  5). 

20.  3  4/(5  +  a;)  +  5  y/{x  -  7)  and  3  |/(5  -\-  x) - 
5  |/(a;  -  7). 

21.  4  |/(7  +  a:)  -  3  ^/(a;  -  4)  and  4  |/(7  +  a;)  + 
3  ^/(^  -  4). 

3i/(2a;  +  4)  3a:-  10  ^    ^,„      ,    ,, 

*  Unless  otherwise  stated,  directiojis  for  I  apply  to  11  also. 


114  MULTIPLICATION  AT  SIGHT. 

23.  A  cistern  could  be  filled  by  one  pipe  alone  in  six 
hours,  and  by  another  pipe  alone  in  eight  hours ;  and  it 
could  be  emptied  by  an  outlet  pipe  in  twelve  hours.  In  how 
many  hours  would  the  cistern  be  filled  were  all  three  pipes 
opened  together  when  the  cistern  was  empty  ? 

94.  Product  oi  x  -\-  y  and  x^  —  xy  -\-  y"^. — The  product 
oi  X  -\-  y  and  x^  —  xy  +  ?/^  is  x^  +  y^,  and  of  x  —  y  and 
x^  -{-  xy  -\-  y^  is  x^  —  y^.  (Show  these  by  actual  multiplica- 
tion. ) 

In  words,  the  product  of  the  sum  of  two  terms  and  the 
sum  of  the  squares  of  the  terms  minus  their  product  is  the 
sum  of  the  cubes  of  the  terms,  and  the  product  of  the  dif- 
ference of  two  terms  and  the  sum  of  the  squares  of  the 
terms  plus  their  product  is  the  difference  of  the  cubes  of 
the  terms. 

EXERCISE  XLVIII. 

Write  at  sight  the  product  of  the  following  pairs  of 
factors : 

I. 

1.  X  -\-  a  and  a^  —  ax  -\-  a\ 

2.  X  -}-  3  and  x^  —  Sx  -\-  9. 

3.  X  —  7  and  x^  -{-  7x -\-  49. 

4.  X  —  c  and  x^  -\-  ex  -\-  c^. 

6.     2a;2  -  3«  and  4.x^  +  Qax^ .+  9a^. 

Write  at  sight  the  missing  factor  of  the  two  following 
examples : 

6.  (x  -  4)(  )=x^  -  64. 

7.  (2aa;2  +  7)(  )  =  S^s^e  _^  343^ 

8.  Square  xf  -\-  x^  -\-  x  -\-  1  by  the  method  of  section 
73. 

9.  Cube  1  —  dx^  -\-  2x*  by  the  method  of  section  75. 


MULTIPLICATION  AT  SIGHT.  116 

II. 

10.  a^  +  1/db  and  a^  -  l/3a^  +  l/9b\ 

11.  1/2 A3  _  2/3^>V  and  l/4aV  +  \/Za%^o?  + 
4/9&V. 

12.  1/5 «V  +  1/6^V    and    l/'lha^'x^  -  l/ZWl^x^  + 

l/36^»Vo. 

Write  at  sight  the  missing  factor  of  the  following 
examples : 

13.  (3«V  -  l/3aa;)(  )  =  27A»  -  X/Vta^^, 

14.  (1/46^32:5  +  1/6^V)(  )  =  1/64A15  + 
l/216J«a^2i_ 

95.  To  Convert  x^  +  Zia;  into  a  Perfect  Square. — The 

square  of  a  binomial  of  the  first  degree  of  the  form  x  -\-  a, 
that  is,  of  one  having  a  constant  term  and  unity  as  the  co- 
efficient of  its  first-degree  term,  is  a  complete  quadratic 
trinomial.  The  first-degree  term  of  this  trinomial  is  twice 
the  product  of  the  two  terms  of  the  binomial,  and  the  con- 
stant term  of  the  trinomial  is  the  square  of  the  constant 
term  of  the  binomial,  or  the  square  of  half  the  coefficient 
of  the  first-degree  term  of  the  trinomial. 

e.g.     {x  +  4)2  =  a;2  +  8a;  +  16.     Here  16  =  (|)l 

{x  -  1/2)2  ^x^  -x-^  1/4.    Here  1/4  =  (1/2)2. 

Hence  a  quadratic  binomial  of  the  form  x^  +  ^^?  that 
is,  one  having  a  first-  and  a  second-degree  term  in  a  letter 
and  unity  as  the  coefficient  of  its  second-degree  term,  may 
be  converted  into  a  perfect  square  by  adding  as  a  constant 
term  the  square  of  half  the  coefficient  of  its  first-degree 
term. 

e.g.  The  quadratic  binomial  :i^  —  ^x  becomes  a  perfect 
square  on  the  addition  of  (3)2  to  it  as  a  constant  term. 
When  thus  completed  it  becomes  the  trinomial  x?'—^x-\-^. 


116  MULTIPLICATION  AT  SIGHT. 

EXERCISE  XLIX. 

Convert  the  following  quadratic  binomials  into  perfect 
squares : 

I. 


1. 

a?  +  8:^;. 

2. 

m2  -  10m. 

3. 

a^-^x. 

4. 

n^  —  hn. 

6. 

x^  +  lx. 

6. 

II. 

f  -  %. 

7. 

x^  -  Z/^x, 

8. 

z^  +  h/^z. 

9. 

x^  +  Ix. 

10. 

x^  -  6bx. 

11. 

a?-{-x. 

12. 

y'-y- 

96.  To  Convert  x^  +  bx""  into  a  Perfect  Square.— Bi- 
nomials of  a  similar  form  but  of  a  higher  degree  may  be 
converted  into  perfect  squares  in  the  same  way.  The  form 
of  the  expression  will  be  similar  when  the  degree  of  one 
term  in  any  letter  is  twice  that  of  the  other  term  in  the 
same  letter,  and  the  coefficient  of  the  term  of  the  higher 
degree  is  unity. 

e.g.  x^  —  8x^  becomes  a  perfect  square  on  the  addition 
of  (4)2.  It  will  then  be  x^  -  Sx^  +  16.  This  is  the  square 
of  x^  —  4. 

Of  course  in  any  of  these  cases  an  aggregate  may  take 
the  place  of  a  single  literal  factor.' 


EXERCISE  L. 

Convert  the  following 
plete  squares: 

binomial  expressions  into  com- 
I. 

1.     x^  +  6x^ 

2.    m*  -  12m\ 

3.     x'  -  5x^. 

4.     a^  +  7a\ 

6.    x'-\-bx^ 

MULTIPLICATION  AT  SIGHT.  117 

II. 

6.     z^  -  z^.  7.     ^^«  -  2/3:?:^ 

8.     71^  -  3/4^^3.  9.     {x  +  2)2  +  Q{x  +  2). 

10.     {x  -  5)2  -  '6{x  -  5). 

97.  To  Convert  x?  -\- bx  -\-  c  into  a  Perfect  Square. — 
Quadratic  trinomials  of  the  form  x^  -{-  hx  -\-  c  may  be  con- 
verted, without  change  of  value,  into  perfect  squares  plus 
or  minus  a  term  which  may  be  either  simple  or  complex, 
by  the  addition  and  subtraction  of  the  square  of  half  the 
coefficient  of  x.  It  is  best  to  make  the  addition  and  sub- 
traction immediately  after  the  second  term,  and  then  to 
combine  the  last  two  terms  into  one. 

e.g.    cc2  +  4a;  —  8  =  a;2  -f-  4a:  -|-  4  —  4  —  8 
=  a;2  _j_  4^  _|_  4  _  12. 

The  first  three  terms  of  the  last  polynomial  are  a  perfect 
square. 

x^-^^x-\-10  =  x^-\r  6x  +  9-9  +  10  =  a;^  +  6a:  +  9  + 1. 

x^J^bx-    7=za:2+5a;  +  ?^-^-7 

4  4 

=  ^'  +  5a;  +  —  -  — . 

EXERCISE  LI. 

Convert  each  of  the  following  trinomials  into  a  perfect 
square  plus  or  minus  a  constant  term,  without  change  of 
value : 

I. 

1.     a;2  -  8a;  -  2.  2.     x^  -  12a;  +  30. 

3.     x^  +  7a;  ^  3/4.  4.     a;^  -  7a;  +  3/5. 

5.  Divide  l/32a;5  -  1024  by  l/2a;  -  4. 


118  MULTIPLICATION  AT  SIGHT. 

6.  A  workman  was  employed  for  60  days,  on  condi- 
tion that  he  should  receive  3  dollars  for  every  day  he 
worked,  and  forfeit  1  dollar  for  every  day  he  was  absent. 
At  the  end  of  the  time  he  received  48  dollars.  How  many 
days  did  he  work  ? 

7.  A  can  do  a  piece  of  work  in  10  days,  and  B  can  do 
it  in  eight  days.  After  A  has  been  at  work  on  it  for  three 
days,  B  comes  to  help  him.  In  how  many  days  will  they 
finish  ? 

II. 

8.  /  -  %  +  3.  9.     z^  +  Hz  -  7. 
10.     x^  -\-'bx  -\-  c.                  11.     y^  —  hy  —  c. 

12.  Divide  32/243a;5  +  3125  by  2/3:c  +  5. 

13.  A  privateer,  running  at  the  rate  of  10  miles  an 
hour,  discovers  a  ship  18  miles  off  running  at  the  rate  of  8 
miles  an  hour.  How  many  miles  can  the  ship  run  before 
she  is  overtaken  ? 

14.  A  cistern  has  two  supply-pipes  respectively  capable 
of  filling  it  in  4|  and  6  hours.  It  also  has  a  leak  capable 
of  emptying  it  in  5  hours.  In  how  many  hours  would  it  be 
filled  when  both  pipes  are  on  ? 

98.  To  Convert  ax^  -j-  hx  into  a   Perfect  Square.  — 

Quadratic  binomials  of  the  form  ax^  -\-  hx  may  be  converted 
into  perfect  squares  by  first  dividing  them  by  the  coefficient 
of  x^  and  then  adding  the  square  of  half  of  the  resulting 
coefficient  of  x. 

e.g.  Zx^  +  12a;  becomes,  on  division  by  3,  x^  +  4a:,  and 
then,  on  addition  of  the  square  of  half  of  4,  x^  -\-  ^x  -{-  4, 
which  is  a  perfect  square. 

Similarly,  ^x^  —  5x  becomes  x^  —  6/Sx,  and  then  x"^  — 
5/3x  -f  25/36,  which  last  is  a  perfect  square. 


MULTIPLICATION  AT  SIGHT.  119 

EXERCISE  Lll. 

Convert  the  following  quadratic  binomials  into  perfect 
squares,  and  solve  the  given  equations : 

I. 
1.     Qx^  +  l%x.  2      Sa^  -  15a;. 

3.     6x^  -  15a;.  4.     7a:2  _|_  63^^ 

6.  d{x-^aY-5(x^a).         6.     ^^ZT^^^^^' 

II. 

7.  ax^  +  ix.  8.     my^  —  ?i«/. 
9.     2x*  +  3a;2.                              10      ^sz^  _  9;23 

a.  7(.-5)^+3(.-5)^.  X..  :-±i=:-^. 

EXERCISE  Llil. 

Convert  the  following  quadratic  trinomials,  without 
change  of  value,  into  expressions  which  shall  be  a  perfect 
square  plus  or  minus  a  constant  term : 

I. 

1.  2a;2  -f  32;  +  6.  2.    Sx^  -  18a;  -  12. 

3.  4a;2  -Qx  +  7.  4.     5a;2  +  25a;  -  20. 

5.  6a;2  +  42a;  +  50. 

6.  Find  the  square  root  of  2  to  four  places  of  deci- 
mals. 

7.  Find  the  cube  root  of  3  to  three  places  of  decimals. 

II. 

8.  7a;2  -  63a;  +  49.             9.  Sa^  -  40a;  -  12. 
10.     9ar^  -  81a;  +  63.           n.  lOa;^  _^  70a;  -  80. 
12.     lla;^  —  2a;  +  3.              13.  ax^  -\-  bx  -\-  c. 

14.     mz^  —  7iz  +  p. 


CHAPTER  XL 
PACTORING. 

99.  Resolution  into  Factors. — To  factor  an  expression 
is  to  resolve  it  into  its  component  factors.  To  be  able  to 
factor  algebraic  expressions  readily  and  accurately  is  a  mat- 
ter of  very  great  importance.  Other  things  being  equal, 
the  one  most  skilful  at  factoring  is  the  best  algebraist. 

1°.  To  Resolve  an  Expression  into  a  Monomial  and  a 
Polynomial  Factor. — When  every  term  of  a  polynomial 
contains  a  common  factor,  it  may  be  resolved  into  a  mono- 
mial and  a  polynomial  factor. 

The  factor  common  to  all  the  terms  will  be  the  mono- 
mial factor,  and  the  quotient  obtained  by  dividing  the  ex- 
pression by  this  factor  will  be  the  polynomial  factor. 

e.g.         6:^2  _^  i2x  -  18  =  Q{x^  +  2^;  -  3). 

a^x  —  a^  =  a^(x  —  1). 

EXERCISE  LIV. 

Eesolve  each  of  the  following  expressions  into  a  mono- 
mial and  a  polynomial  factor : 

I. 

1.     Gab  +  2ac,  2.    2a^x^  -  %d^hx  +  Wh\ 

3.     ^l)^(?x  +  bbh^y  -  6b^c\    4.    7a  -  7a^  +  UaK 

5.     Qx^  +  2x^  +  4:X^. 

120 


FAGTOniNQ. 

II. 

T. 

bu^- 

-  lOA^  - 

9. 

^^- 

-x^^x. 

121 


6.  15a2  -  225^^  T.    ^x^  -  lOA^  -  15aV. 
8.     38«V  +  57aV. 

10.     '^x^y^  —  'Sx^y^  -\-  2xy^. 

2°.  To  Facto?'  the  Difference  of  Two  Squares. — The 
difference  of  two  squares  is  equal  to  the  product  of  the  sum 
and  difference  of  their  roots. 

EXERCISE   LV. 

Factor  each  of  the  following  expressions : 
I. 
1.     x^  —  a^.  2.     x^  —  9. 

3.     4^2  -  64.  4.     9  A2  _  25^2, 

5.     81  -  16A*.  6.     49fi^V  -  lQa^z\ 

7.  {x^  +  Ux  +  36)  -  49.      8.     y^-8y  +  16  -  81. 

9^     («2  _  4rt  +  4)  _  16.  10.     (^2  ^  24.b  +  144)  -  121. 

11.  The  head  of  a  fish  is  9  inches  long,  the  tail  is 
as  long  as  the  head  and  half  the  body,  and  the  body  is  as 
long  as  the  head  and  tail  together.  What  is  the  length  of 
the  fish? 

Note. — In  solving  problems  concerning  numbers  composed  of 
digits,  the  student  must  bear  in  mind  that  a  number  composed  of  two 
digits  is  equal  to  10  times  the  left-hand  digit  plus  the  right-hand 
digit;  that  a  number  composed  of  three  digits  is  equal  to  100  times 
the  left-hand  digit  plus  10  times  the  middle  digit  plus  the  right-hand 
digit.     Thus,  46  =  10  X  4  +  6,     and    387  =  100  X  3  +  10  X  8  +  7. 

12.  A  number  is  composed  of  two  digits,  and  the  left 
digit  is  4/3  of  the  right.  If  18  be  subtracted  from  the 
number,  its  digits  will  be  reversed.     What  is  the  number  ? 

II. 

Factor : 

13.  12  -  3a2.  14.     48«3  _  lOSabK 


122  FAGTORINO. 

15.     27«5  -  75«.'r^  16.     125«V  _  45:ry. 

Convert  the  following  trinomials  into  the  difference  of 
two  squares  and  then  factor: 

17.     0?  +  14a:  +  40.  18.     x^  -  l^x  -  17. 

19.     x^  -  lOx  -  11.  20.     x^  +  30x  +  29. 

21.  A  and  B  together  can  do  a  piece  of  work  in  12 
hours,  A  and  0  together  can  do  it  in  16  hours,  and  A 
alone  can  do  it  in  20  hours.  In  what  time  can  they  all  do 
it  together,  and  in  what  time  could  B  and  C  together  do 
it? 

22.  A  number  is  composed  of  two  digits  whose  sum  is 
13,  and  if  9  be  added  to  the  number  its  digits  will  be 
reversed.     What  is  the  number  ? 

3°.  Special  Cases  of  Factoring  Quadratic  Trinomials. 
— We  have  seen  that  the  product  of  two  binomials  of  the 
first  degree  in  any  letter  is,  in  general,  a  quadratic  trino- 
mial in  the  same  letter,  and  that  the  coefficient  of  the  sec- 
ond-degree term  of  the  letter  is  the  product  of  the  coeffi- 
cient of  the  first-degree  terms  of  the  letter  in  the  binomials, 
the  coefficient  of  the  first-degree  term  of  the  letter  in  the 
product  is  the  sum  of  the  products  of  the  coefficient  of  the 
first-degree  term  of  the  letter  in  each  binomial  multiplied 
by  the  constant  term  of  the  other  binomial,  and  the  con- 
stant term  of  the  product  is  the  product  of  the  constant 
terms  of  the  binomials. 

Hence  a  quadratic  trinomial  in  any  letter  may  be  re- 
solved into  two  binomial  factors  of  the  first  degree  in  that 
letter  whenever  we  can  discover  four  numbers  such  that  the 
product  of  the  first  two  will  be  the  coefficient  of  the  second- 
degree  term  of  the  trinomial,  the  product  of  the  last  two 
will  be  the  constant  term  of  the  trinomial,  and  the  alge- 
braic sum  of  the  cross-products  of  the  numbers  will  be  the 


FACTORING. 


123 


coefficient  of  the  first-degree  term  of  the  trinomials.  The 
first  two  numbers  will  then  be  the  coefficients  of  the  first- 
degree  terms  of  the  factors,  and  the  last  two  numbers  will 
be  the  constant  terms  of  the  factors. 

It  is  best  to  arrange  diagrammatically  the  four  numbers 
selected  for  trial,  as  in  the  corresponding  case  of  sight  mul- 
tiplication. 

e.g.  Resolve  Qx^  +  '^•^  ~  ^0  into  binomial  factors. 
3x2  =  6,  the  coefficient  of  x^; 

2  X  -4^-8;  ^' 

3  X  5  =  15; 
15  +  (—  8)  =  7,  the  coefficient  of  x; 


5  X 


the  constant 


(_  4)  =  -  20, 
term. 
Hence  ^x'^  +  7:?;  -  20  =  (2^  +  5)(3a;  -  4). 
Notice  that  the  complete  test  involves  two  trials,   if 
first  be  unsuccessful:    e.g.   3  above 
'^ '  and  2  below  as  well  as  2  above  and  3 
below. 

Again,    resolve   3a;^  —  l^x  —  63 
into  binomial  factors. 

The  required  factors  are  {x  —  7) 
and  {3x  +  9). 
Resolve  x'^  —  2x  —  63  into  bino- 
mial factors. 

The  factors  are  {x  -\-  7)  and 
(X  -  9). 

The  case  in  which  the  coefficient 
of  the  second-degree  term  of  the  tri- 
nomial is  unity  is  of  frequent  occur- 
rence and  of  great  importance. 


124  FAGTOBING. 


EXERCISE  LVI, 

Eesolve  the  following  quadratic  trinomials  into  binomial 
tors.- 

I. 

1.       X' -^  l^X -\-  ^b.                          2. 

x^  -  12a;  +  27. 

3.     x^  -^x-  32.                    4. 

x^  +  lx-  30. 

5.      X^  —  X  —  42.                          6. 

x^  +  x-  20. 

7.     %x^  -  lOx  -  48.                 8. 

3.^2  _|_  26a;  +  55. 

9.     Qx"  -  17^  +  7.                 10. 

202;^  +  37^  +  8. 

11.     'dbx^  +  39^  -  36.             12. 

56a;2  -  lOOx  -  100. 

13.  A,  B,  and  0  together  can  do  a  piece  of  work  in  5 
days,  A  and  B  together  can  do  it  in  8  days,  and  B  and  0 
together  in  7  days.     In  what  time  can  each  do  it  alone  ? 

II. 

Factor  the  following  expressions : 

14.  12  +  10^  -  %xK         15.     48  -  128^  +  Mx\ 

16.     35  4-  41a;  +  122;2.       17.     6a;2  +  (21  -  ^a)x  -  7«. 

18.  ahx^  -\-  {"Ha  —  5b)x  —  35. 

19.  acx^  +  (^c  —  ad)x  —  M. 

20.  x^^^bx-a^-^y"' 

21.  {d^  -  y^)x^  -  2{a  +  db)x  -  8. 

22.     3a;2  +  9a;  -  54.  23.     7x^  -  7x  -  210. 

24.     102;2  +  50a;  -  140.  26.     75flV  -  5  A  -  30^2. 

26.  Find  a  number  composed  of  two  digits  whose  sum 
is  twelve  and  which  will  have  its  digits  reversed  by  adding 
63  to  the  number  and  dividing  the  sum  by  4. 

100.  Functions. — In  mathematics,  one  quantity  is  said 
to  be  a  function  of  another  when  its  value  depends  upon 
the  value  of  the  other  and  changes  with  it. 


FACTORING.  125 

e.g.  The  value  of  the  expression  x^  -{-  ^x  —  6  depends 
upon  the  value  of  x  and  changes  with  the  value  of  x. 
Hence  the  expression  a:^  -f-  6^  —  6  is  a  function  of  x. 

The  symbol  f{x)  means'  any  algebraic  expression  con- 
taining X.  This  is  a  very  convenient  notation  when  we 
wish  to  indicate  any  expression  containing  x  without  des- 
ignating any  particular  expression.  f{a)  indicates  the 
algebraic  expression  obtained  by  substituting  a  for  x  in 
f{x).     Thus  \if{x)  =  x'^-\-'dx^Q,  then 

f{a)  =  6?2  +  3a  +  6. 

EXERCISE  LVII. 
I. 

1.  lif(x)  =x^-{-  dx^  -  10,  find/(3). 

2.  ltf{x)  ^x^  +  3x^  -  10,  find/(-  3). 

3.  If /(^)  =  x^  —  5x  -\-  Q  and  y  =  3  —  x,  ^ndf{y)  in 
terms  of  x. 

4.  ltf(x)  =x^-irx-Jrl,  find /(a;  -  1). 
6.     Itf{x)  =  x^-]-2x-  7,  find/(5). 

6.  If /(ft)  =  (ft  +  J  +  c)3  -  «3  _  J3  _  ^3^  find/(-^»). 

7.  If /(^)  =  x'-  y\  find  f\y). 

8.  \if{x)  =  x'-y\^^^f{y). 

II. 

9.  \if{x)=x--y\^v.^f{y), 

10.  If/(a:)  =  a;5  +  ^^find/(-^). 

11.  \lf{x)  =  x^^y\^T,^f{y). 

12.  ^/(2;)  =  a:^  +  y^find/(-2/). 

13.  If/(a:)=:a;^-fy*,  find /(«/). 


126  FACTORING. 

14.  lif(x)  =  x""  +  y''  and  n  is  odd,  find/(-  y). 

15.  If /(^)  =  a;"  +  2/"  ^^^  ^  is  even,  find/(—  y). 

16.  If/(a;)  =  a:~  +  ?/%find/(2/). 

101.  Remainder  Theorem. — When  f{x)  is  divided  by 
X  —  a,  the  process  of  division  being  continued  till  the  re- 
mainder, if  there  be  one,  does  not  contain  x,  the  remainder 
will  =/(«). 

Proof. — Denote  the  remainder,  which  is  supposed  not 
to  contain  x,  by  R  and  the  quotient  by  Q.     Then  we  have 


X  —  a  X  —  a 

or  f{x)  ^  Q(x  -  a)-\-  R. 

If  now  we  substitute  a  for  x  in  each  member,  R  must 
remain  unaltered  since  it  does  not  contain  x,  and  x  —  a  will 
become  a  —  a  =  0.     Hence  f{a)  =  R. 

e.g.  Letf(x)  =  x^  -\-  2x^  —  5x  —  6,  and  let  a  =  4. 
Then 

/(«)  =  43  +  2  X  42  -  5  X  4  -  6  =  64  +  32  -  20  -  6=70. 

By  division, 

x^-\-2x^-    6x-    6\x  -4: 


a^-\-6x-\- 19 


6x^  -    5x 
ex^  -  Ux 


19.T  -    6 

l^x  -  76 

70 
Again,  let  f{x)  =  x^  -{-  32,  and  let  a 
Then  f{a)  =  32  +  32  =  64. 


FACTORING.  127 

By  division, 


x> 

+  32 
-    ^x" 

\x  -2 

X' 

x^  +  2x 
f  32 
-    4:c3 

^  +  4a; 

32 

8:rM 
Sx'- 

2  +  8a;  +  16 

2x^- 
2x'- 

4^3  + 

-32 
-  16a: 

16a;  +  32 
16a;  -  32 

Again, 

let 

f{^) 

=  a;5- 

-32, 

and  let  a  - 

=  2. 

Then 

Ao) 

=  32 

-32 

=  0. 

By  division, 

x^- 

32 

\x- 

2 

x"  - 

2x^ 

nA       1 

Oo,3     1 

1     /1/V.2*    1     Q/>. 

1   1 

2a;*  -  32 
2a;*-    4a;'^ 


4a;3  -  32 

4a;3-    Sa;^ 

8a;2  -  32 
8a;2  -  16a; 


16a;  -  32 
16a;  -  32 

The  theorem  proved  and  illustrated  above  is  a  fun- 
damental theorem  in  factoring.  By  it  we  can  readily  de- 
termine whether  x  —  a  is  a  factor  of  /(a;).  We  have 
merely  to  substitute  a  for  x  in  the  given  expression,  and  see 
whether  it  reduces  to  zero  or  not.     In  the  former  case  the 


128  FACTORING. 

expression  is  divisible  hy  x  —  a  without  remainder,  and 
therefore  :r  —  «  is  a  factor  of  it.  In  the  latter  case  the 
expression  is  not  divisible  hy  x  —  a  without  remainder,  and 
therefore  x  —  ai^  not  a  factor  of  it. 

EXERCISE    LVIII. 

Find  in  each  of  the  following  examples  whether  or  not 
the  given  binomial  is  a  factor  of  the  given  expression : 

1.  x-b  of  x^  -  Ix^  +  7a;  +  15. 

2.  X -\- 1  oi  "^x^  -\-  X  —  1. 

3.  X  —  1  of  x^  -\-  x^  —  2. 

4.  a;  -  3  of  'Zx^  +  10a;2  -  ^x  -  40. 

6.  X  —  h  oi  x^  —  h^.  6.     X  -{-!)  of  x''  -\-  V, 

7.  X  -{-h  of  x^  —  W.  8.     X  —  h  of  x^  +  y^, 
9.     X  —  h  olx^  —  b^.         10.     X  -\-  b  of  x^  —  b^. 

11.     x  —  b  of  a;^  +  b^.         12.     is  +  2»  of  x^  +  i^ 

.13.  OJ  —  Z>  of  a;"  +  Z*"  when  w  is  odd. 

14.  X  —  b  of  a;"  +  ^"  when  w  is  even. 

16.  X  -{-  b  of  x''  -{-  ^"  when  ^  is  odd. 

16.  X  -\-  b  of  x^  +  ^"  -when  ?^  is  even. 

17.  X  —  b  of  X"  —  ^"  when  ^  is  odd. 

18.  X  —  b  of  x^  —  J"  when  ^i  is  even 

19.  X  -\-  b  of  ic"  —  &"  when  7i  is  odd. 

20.  X  -\-  b  of  a:"  —  ^"  when  ^  is  even. 

21.  Divide  x*  —  b~  hy  x  —  b. 

22.  Divide  x^  —  b^  hy  x  -\-  b. 

23.  Divide  x^  -^  b^  hy  x  —  b. 

24.  Divide  a;^  +  b^  by  x  -\- b. 


FAC  TORINO.  129 

26.     Divide  x^  -\- h^  ^-^  x  —  h. 

26.  Divide  x^  -\-  IP  hj  x  -\-  h. 

27.  Divide  x'^  —  h^hj  x  —  h. 

28.  Divide  x^  —  Whj  x-\-  h. 

102.  Factors  of  the  Sum  and  Difference  of  the  Same 
Powers  of  Two  Quantities. — From  examples  13-20  it  ap- 
pears : 

1°.  That  the  sum  of  the  same  odd  powers  of  two  quan- 
tities is  divisible  by  the  sum  of  their  roots,  but  not  by  the 
difference  of  their  roots. 

2°.  That  the  sum  of  the  same  even  powers  of  two  quan- 
tities is  divisible  by  neither  the  sum  nor  the  difference  of 
their  roots. 

3°.  That  the  difference  of  the  same  odd  powers  of  two 
quantities  is  divisible  by  the  difference  of  their  roots,  but 
not  by  the  sum  of  their  roots. 

4°.  That  the  difference  of  the  same  even  powers  of  two 
quantities  is  divisible  by  both  the  sum  and  difference  of 
their  roots. 

From  examples  21,  26,  27,  28,  it  appears: 

1°.  That  when  the  difference  of  the  same  powers  is  di- 
vided by  the  difference  of  the  roots,  the  terms  of  the  quo- 
tient are  all  positive ;  and  that  when  the  sum  or  difference 
of  the  same  powers  is  divided  by  the  sum  of  the  roots,  the 
terms  of  the  quotient  are  alternately  positive  and  negative. 

2°.  That  in  any  case  the  first  term  of  the  quotient  is 
the  letter  of  the  first  term  of  the  dividend  with  its  exponent 
diminished  by  one,  and  that  the  exponent  of  this  letter  de- 
creases by  one  in  each  of  the  succeeding  terms  of  the  quo- 
tient; and  that  the  letter  of  the  second  term  of  the  dividend 
occurs  in  the  second  term  of  the  quotient  with  unity  for  its 
exponent,  and  that  the  exponent  of  this  letter  increases 


130  FACTORING. 

by  one  in  each  subsequent  term  till  it  becomes  one  less  than 
its  exponent  in  the  dividend. 

These  two  laws  enable  us  in  these  cases  of  division  to 
write  the  quotient  at  sight. 

EXERCISE  LIX. 

Write  at  sight  the  quotient  in  each  of  the  following 
cases : 

1.  {x^  -  if)  ^  (x-  y).  2.  (x^  -  f)  -^{x-  y). 
3  (^6  _  ^6)  ^(^x^  y).  4.  {x'  +  f)  ^{x  +  y). 
5.  {x^  -  27)  ^{x-  3).  6.  {x^  -  81)  -^{x-  3). 
7.     (.T^  -  16)  ^  (;?:  +  2).  8.     {x?  +  32)  ^  (x  +  2). 

Find  the  remainder  when — 

I. 
9.     {x  —  %af  +  (%x  —  (if  is  divided  by  re  —  «. 

10.  {x -\-  a -\- Hf  ^  x^  is  divided  by  2;  +  a. 

11.  {x  +  2«)2"  +  (2.^  +  of"  -  2«'^"  is  divided  by  a:  +  a. 

12.  («  +  ^  +  ^Y  —  ^'^  —  Ifi  ~  &  is  divided  by  a  +  h. 

II. 

13.  {a  ^h  -\-  cf  —  ci}  —  h~  —  c'  is  divided  hy  a  -\-d. 

14.  {a-^b^cY  -  {b-[-  cy-  (c  +  aY  -  («  +  hy  + 
^^  +  **  +  c*  is  divided  by  a  +  ^. 

15.  «"(^  —  c)+  J"(^  —  «)+  ^"(^  —  ^)  is  divided  by  h—c. 
Show  that  the  given  binomial  is  a  factor  of  each  of  the 

following  expressions,  and  find  the  other  two  factors : 

I. 

16.  3a:3  +  a;2  -  22:2:- 24;  a:- 3. 

17.  x^  +  2.^2  -  13a:  +  10;  x  -  2. 

18.  x^-]-%x'^  -  11a;  -  12;  ;r  +  1. 


FACTORING.  131 

II. 

19.  3a;'  -  %0x^  +  36:c  -  16;  x-  L 

20.  ^x^  +  13a;2  -  32.i-  +  15 ;  x  +  5. 

EXERCISE  LX. 

I. 

1.  A  cistern  can  be  filled  by  one  pipe  in  five  hours  and 
by  another  in  eight  hours,  and  it  can  be  emptied  by  a  third 
pipe  in  four  hours.  Were  the  cistern  empty  and  all  three 
pipes  opened  together,  in  what  time  would  it  be  filled  ? 

2.  Suppose  the  cistern  in  the  last  example  could  be 
emptied  by  the  third  pipe  in  three  hours.  Were  the  cistern 
full  and  all  three  pipes  opened  together,  in  what  time  would 
it  be  emptied  ? 

3.  A  man  does  3/5  of  a  piece  of  work  in  30  days  and 
then  calls  in  another  man  and  they  together  finish  it  in  6 
days.     In  what  time  can  they  do  it  separately  ? 

II. 

4.  A  marketwoman  bought  a  number  of  eggs  at  the 
rate  of  two  for  a  penny,  and  as  many  more  at  the  rate  of 
three  for  a  penny,  and  sold  the  whole  at  the  rate  of  four 
for  3  cents,  and  found  she  had  made  24  cents.  How  many 
of  each  kind  did  she  buy  ? 

5.  A  person  hired  a  laborer  on  condition  that  he  was 
to  receive  2  dollars  for  every  day  he  worked  and  forfeit 
75  cents  for  every  day  he  was  absent.  He  worked  three 
times  as  many  days  as  he  was  absent,  and  received  $47.25. 
How  many  days  did  he  work  ? 

6.  A  sum  of  money  was  divided  between  A  and  B,  so 
that  the  share  of  A  was  to  that  of  B  as  5  to  4.  The  share 
of  A  exceeded  5/11  of  the  whole  by  300  dollars.  What  was 
each  man's  share  ? 


CHAPTER  XII. 
HIGHEST  COMMON  FACTORS. 

103.  Highest  Common  Factor.  —  A  common  factor  of 
two  or  more  expressions  is  a  factor  which  is  contained  in 
each  of  them,  and  the  highest  common  factor  of  the  expres- 
sions is  the  product  of  all  their  common  factors.  Thus, 
Stt^Z'V  and  Qa^Vc  have  2,  a^,  W,  and  c  as  common  factors, 
and  2«^J^c  as  their  highest  common  factor. 

The  abbreviation  H.  C.  F.  stands  for  highest  common 
factor. 

The  highest  common  factor  is  sometimes  called  the 
greatest  common  measure,  and  denoted  by  G.  C.  M. 

104.  The  H.  C.  F.  of  monomials  may  be  found  by  in- 
spection. It  is  necessary  merely  to  factor  the  expression, 
select  the  common  factors  and  find  their  product,  using 
each  of  these  factors  the  least  number  of  times  that  it 
occurs  in  any  of  the  expressions. 

e.g.  Find  the  H.C.F.  of  \WIHH,  9«^^>V,  and  Vla^WdK 
Factoring,  we  have 

Z.'^.^.a.a.h.h.'b.c.c.c.c.d^ 

S.d.a.a.a.b.b.c.c.c.c.c, 

and  Z  .  %  .  %  .  a  .  a  .  a  .  a  .  h  .  h  .  h  .  h  .  d  .  d .  d .  d. 

The  factors  common  to  all  the  expressions  are,  3,  a, 
and  J).  The  least  number  of  times  that  3  occurs  in  any  of 
the  expressions  is  oncej  that  a  occurs  in  any  of  the  expres- 

189 


niQHEST  COMMON  FACTORS.  133 

sions  is  twice;  and  that  b  occurs  in  any  of  the  expressions 
is  twice.  Now  3  .  a  .  a  .  b  .  b  —  3a^^,  and  this  is  the 
highest  common  factor  of  the  expressions.  Of  course  we 
might  have  seen  at  once  that  the  highest  common  factor  of 
the  coefficients  is  3,  that  the  common  letters  are  a  and  b, 
and  that  the  lowest  dimension  of  these  letters  in  any  of  the 
expressions  is  2.     Hence  the  H.  C.  F.  would  be  da^b"^. 

EXERCISE  LXI. 

Find  the  H.  0.  F.  of  the  following  expressions: 

I. 
1.    5x^1/,  Ibx^ii^z.  2.     7x^yh,  2Sx^yh^. 

3.     lSab^c%  36a^cd\         4.     2xY,  'dxY,  4xY:f. 

6.  ITa^^V,  51a''b^c\  QSa^bh\ 

II. 

7.  Multiply  ^x"^  +  6a:"  -  5xPy^  by  3cc"  -  4:X^  +  Qxy^. 

8.  Divide  6a;"*  +  ^  -f  9a;'"  +  ^  +  12a;"  +  ^  -\-  18a;"  +  ^  —  8x^ 
-  12x^  by  2a;2  +  3a;. 

105.  To  Find  Highest  Common  Polynomial  Factor  by 
Inspection. — In  a  similar  way  we  may  find  the  H.  C.  F.  of 
two  or  more  polynomial  expressions  by  inspection  when  we 
are  able  to  resolve  them  into  polynomial  factors.  We  have 
simply  to  resolve  the  expressions  into  their  polynomial  fac- 
tors, select  the  factors  common  to  all  the  expressions,  and 
combine  them  into  a  product,  using  each  factor  the  least 
number  of  times  that  it  occurs  in  any  of  the  expressions. 

e.g.  Find  the  H.  C.  F.  of  x^ -{- x  -  6,  x^ -\- Qx  +  9,  and 
a;^  —  a;  —  12. 

Factoring,  we  obtain 

{x  4-  3)(a;  -  2),  {x  +  3)(a;  +  3),  and  {x  +  3)(a;  -  4). 


134  HIOHEST  COMMON  FACTOHS. 

The  only  common  factor  is  x  -\-  3,  and  the  least  number 
of  times  that  this  occurs  in  any  of  these  expressions  is  once. 
Hence  the  H.  C.  F.  of  these  three  expressions  is  a;  -f-  3. 

When  any  of  the  polynomial  expressions  contains  a 
monomial  factor,  this  factor  should  be  removed  before 
searching  for  polynomial  factors ;  and  if  this  factor  is  com- 
mon to  all  the  expressions,  or  contains  a  factor  common  to 
them,  the  common  factor  should  be  set  aside  to  be  made  a 
factor  of  the  H.  C.  F. 

e.g.  Find  the  H.  C.  F.  of  3rtV  +  3A  -  60^2,  Qa^x^  - 
d6a\  and  Ua^x^  -  lOSa^x  -f  240«^J. 

Removing  the  monomial  factors,  we  have 

3a%x^  +  x-  20),  6a%x^  -  16),  and  na^{x^  -  9x -\-  20). 

Sa^  is  the  H.  C.  F.  of  the  monomial  factors  thus  re- 
moved. Factoring  now  the  three  polynomial  expressions, 
we  have 

(x  -  4:)(x  +  5),  (x  -  4:){x  +  4),  and  {x  -  4.){x  -  5), 

the  highest  common  factor  of  which  is  2:  —  4.     Therefore 
the  H.  C.  F.  of  the  three  given  expressions  is 

3«2(a:  -  4)  =  3alT  -  na\ 

EXERCISE  LXil. 

Find  the  H.  0.  F.  of  the  following  expressions: 
I. 

1.  x^-l,x^-\-3x-{-2. 

2.  x^-]-5x-\-  6,  x^-{-7x-{-  12. 

3.  x^  -9x-  10,  x^-\-2x-  120. 

4.  x^-^Hx-  18,  x^  -  8. 

6.     x^  -\-  {a  -\-  I?)x  +  CL^,  x^  -\-  {a  —  b)x  —  ab. 

6.  x^  —  Ixy  -\-  6?/^,  x^  —  xy^. 

7.  x^  —  X,  2a:''^  —  4:X  -\-  ^,  x?  -\-  x^  —  2x. 


HIGHEST  COMMON  FACTORS.  135 

II. 

8,  or^  +  y^,  {x  +  yY-,  x^  +  2:r^y  +  '^xy'^  +  2/"- 

9.  120^  +  ly,  6(.7-^  -  1)3,  18(:r  +  1)4. 

10.  2^  -  f,  3(.T-^  -  /),  7(./-«  -  /). 

11.  x^  —  3A  —  2rt^  x^  —  'dax^  +  4«^  x^  —  ax  —  2a^. 

12.  2:^  +  a:?/  —  2;^^,  x^  —  3.^1?/^  +  2?/^,  x^  -\-  3x^y  —  iy'^. 

106.  The  method  of  finding  the  highest  common  factor 
of  two  or  more  expressions  which  cannot  readily  be  resolved 
into  factors  is  based  on  the  three  following  theorems: 

1°.  If  tic 0  exjn'essions  have  a  co?nmon  factor,  any  mul- 
tiples of  these  expressions  will  contain  this  factor. 

Let  A  and  B  represent  any  two  expressions  which  have 
a  common  factor,  and  let  this  factor  be  represented  by  /; 
let  p  denote  the  quotient  resulting  from  dividing  A  by  /, 
and  q  the  quotient  obtained  by  dividing  B  by/.  Then 
A  —  pf  and  B  —  qf.  Let  m  and  n  be  any  integral  expres- 
sions whatever.  Then  mA  will  represent  any  multiple 
whatever  of  A,  and  nB  any  multiple  of  B. 

But  7nA  —  mpf  and  uB  —  nqf. 

Hence  /  is  a  factor  of  both  mA  and  nB. 

2°.  If  two  expressions  have  a  co7nmon  factor,  the  sum 
and  difference  of  the  expressio7is  or  of  any  multiples  of  the 
expressions  will  contain  this  factor. 

Use  the  letters  as  in  1°.     Then 

A  —  B  =  pf  —  qf  =  (p  —  q)f  which  contains  the  factor/. 

Also 
A  -\-  B  =  pf  -\-  qf  =  {p  -{-  q)f,  which  contains  the  factor/ 

Again,  niA  —  ?iB  —  mpf  —  nqf  —  (mp  —  nq)f,  which 
contains  the  factor  / 

Also  mA  -{-  nB  =  mpf  +  nqf  =  (mp  -\-  nq)f,  which 
contains  the  factor/. 


136  HIGHEST  COMMON  FACT0B8. 

3°.  If  two  expressions  have  a  cornmon  factor,  and  one 
of  them  be  divided  by  the  other  and  there  be  a  remainder, 
this  remaitider  will  contain  the  common  factor. 

Let  A  and  B  represent  the  two  expressions  which  have 
a  common  factor,  Q  the  quotient  obtained  by  dividing  B  by 
A,  and  E  the  remainder.     Then 

B=  QA-^  E. 

By  hypothesis  B  and  A  have  a  common  factor  /,  and  by 
1°,  QA  contains /as  a  factor.  But  since  B  is  divisible  by 
/,  and  one  term  of  its  equivalent  expression  (QA  -j-  E)  is 
divisible  by  f  the  other  must  be  also.  Hence  the  remainder 
E  must  contain  /  as  a  factor. 

OoK. — If  now  we  divide  A  hy  E  and  denote  the  remain- 
der by  S,  then  the  common  factor  of  E  and  S  will  be  the 
same  as  that  of  A  and  E  and,  therefore,  of  A  and  B. 

If  this  process  be  continued  to  any  extent,  the  common 
factor  of  any  divisor  and  the  corresponding  dividend  will 
be  a  common  factor  of  the  original  expressions.  In  other 
words,  the  remainder  ivill  always  contain  the  common  fac- 
tors of  the  original  expressions. 

If  at  any  stage  there  is  no  remainder,  the  divisor  must 
be  a  factor  of  the  corresponding  dividend,  and  therefore, 
since  it  is  evidently  the  highest-factor  of  itself,  it  must  be 
the  H.  C.  F.  of  the  original  expressions. 

By  the  nature  of  division  the  remainders  are  necessarily 
of  lower  and  lower  dimensions,  and  hence,  unless  at  some 
stage  the  division  leaves  no  remainder,  we  must  ultimately 
reach  a  remainder  which  does  not  contain  the  common  let- 
ter.    In  this  case  the  given  expressions  have  no  H.  C.  F. 

As  the  process  we  are  considering  is  to  be  used  only  to 
find  the  highest  common  polynomial  factor,  it  is  evident 
that  any  dividend  or  divisor  which  may  occur  in  the  pro- 
cess may  be  multiplied  or  divided  by  any  monomial  factor 
without  destroying  the  validity  of  the  operation;  for  such 


HIGHEST  COMMON  FAG  TOM.  13? 

multiplication  or  division  will  not  affect  the  polynomial 
factors. 

Ex.  1.   Find  the  H.  C.  F.  of 

^3  _|_  ^2  _  2  and  x^  +  2a:2  -  3. 


Q^  +  2a;2  -  3 

\x^  +  x^-% 

a;3  +    a;2  -  2    ' 
a;3  _|_  ^2  _  2           x^  -1 

1 

^^  -  ^                   ^  +  1 
x^  +  x-  2 
^      -      1 

x^-1        x-1 

^'-^       ^4-1 

x-1 

x-1                   The  H. 

C.  F.  is  a;  - 

The  work  might  be  shortened  by  noticing  that  the  fac- 
tors  of  the  first  remainder,  x^  —  1,  are  x  —  1  and  x -\- 1,  and 
that  of  these^  only  a;  —  1  is  a  factor  of  x^  -\-  x^  —  2. 

Ex.  2.  Find  the  H.  C.  F.  of 

x^  +  4.x^y  -  Sxy^  +  'Mif  and  4:X^  -  4cxhj  +  d^xY-^'^xY- 

The  second  expression  is  divisible  by  4:X^,  which  is  evi- 
dently not  a  common  factor.  We  have  therefore  to  find  the 
H.  C.  F.  of  x^  —  x^y  -\-  Sxy^  —  8?/*  and  the  first  expression. 

X*  -    x^y  +  '^xy^  -    %y^     |  a;^  -[-  4:X^y  -  ^xy'^  -\- 
a;4  j^  4^3y  _  8^2^2  ^  24a;^^  V^ITy 

-  hx^y  +    ^xY  -  ^^xy^  -      Sy^ 

-  bx^y  -  %OxY  +  40:r«/3  _  noif 

^SxY  -  562;«/3-f  112^4 


138  HIGHEST  COMMON  FACTORS. 

Rejecting  the  factor  28^/^,  we  have 

rj.  j^  4^^y  _    8:?:/  +  24?/3  |  a.^  -  2xy  +  4:2/ 

^x^y  -  12.t/  _^  24?/3 
6:?;^^?/  —  1  ')lxy'^  +  24?/ 

Hence  the  H.  C.  F.  \%  x^  —  "^xy  -\-  4y^. 
Ex.  3.  Find  the  H.  C.  F.  of 

To  avoid  fractional  coefficients,  the  second  expression 
may  be  multiplied  by  2  and  then  divided  by  the  first. 

Za^+lbx^-^r    bx^^lOx-^'^  12^:4  +  9.^3+14:^  +  3 

2  V, ■ 


Qx^  +  30a:3  _^  iQ^.2  _^  20a:  +  4 
62;4  +  27a:3  +  42a;  +  9 

%x!^  +  9a:3  +  14a:  +  3  |  3^:^  +  lOa;^  _  22^;  -  5 
3 


2a:  +7 


6.T*+27a:3+42a:  +  9 
6a:4+20a:3-44a;2-10a; 

7a:3+44a:2+    52a:- +9 
3 

21a:=^  +  132a:2  +  156a:  +  27 
•       21a:3  +    70a;2  _  154^  _  35 

62a:2  +  310a:  +  62  |  62 
3a:3  +  10a:2  -  22a;  -  5    |     x^ -\-      5a:  +    1 
3a:3  +  15a:^+    3a:  3^  _^ 

—  5a:^  —  25a:  —  5 

-  5a:2  _  25a:  -  5  The  H.  C.  F.  is  a:^  +  5a:  +  1. 

From  the  above  theorems  and  examples  we  may  derive 


HIGHEST  COMMON  FACTORS.  139 

the  following  rule  for  finding  the  H.  C.  F.  of  two  expres- 
sions : 

Arrange  the  two  expressions  according  to  the  descending 
potvers  of  some  common  letter  and,  if  the  expressions  are  oj 
the  same  degree  i7i  that  letter,  divide  either  hy  the  other, 
hit  if  they  are  of  different  degrees  in  that  letter,  divide  the 
one  ivhich  is  of  the  higher  degree  by  the  other.  Take  the 
remainder  after  division,  if  any,  for  a  new  divisor,  and  the 
former  divisor  as  dividend;  and  continue  the  process  till 
there  is  no  re7nai7ider.  The  last  divisor  ivill  be  the  H.  C.  F. 
required. 

If  the  two  expressions  contain  common  monomial  fac- 
tors, their  H.  C.  F.  must  be  obtained  by  inspection,  and  this 
must  be  multiplied  by  the  last  divisor  found  by  the  above 
rule. 

Any  divisor,  dividend,  or  remainder  which  occurs  may 
be  multiplied  or  divided  by  any  monomial  factor. 

107.  To  find  the  H.  C.  F.  of  three  or  more  polynomial 
expressions,  we  first  find  the  H.  C.  F.  of  any  two  of  them, 
and  then  of  this  and  a  third,  and  so  on. 

Let  the  expressions  be  J,  B,  C,  D,  etc. 

First  find  the  H.  C.  F.  of  A  and  B,  and  denote  it  by^. 
Then  since  the  required  H.  C.  F.  is  a  common  factor  of  A 
and  B,  it  must  be  a  factor  of  E,  which  contains  every  com- 
mon factor  of  A  and  B,  and  so  on. 

108.  Note. — The  highest  common  factor  of  algebraic  ex- 
pressions is  not  necessarily  their  greatest  common  measure. 
For  if  one  expression  is  of  higher  dimensions  than  another 
in  a  particular  letter,  it  does  not  follow  that  it  is  numeric- 
ally greater.  In  fact,  if  a  be  a  positive  fraction,  c?  is  less 
than  a. 


140  HWHEBT  COMMON  FACT0M8. 

EXERCISE  LXIII. 

Find  the  H.  C.  F.  of— 

I. 

1.  x^-{-2x-{-l  and  x^  +  2x^  +  2^;  +  1. 

2.  x^  -  8x^  +  7:r  +  24  and  x^  -  6x^  +  8a;  -  6, 

3.  x^  -  5x^  +  dx  +  6  and  x^  -  dx^  -\- ix  -  4=, 

4.  2x^  —  7x-2  and  6x^  -  3x^  -  ISa:^ 

5.  4:X^  +  Sx^  -  6Qx^  -  12a;3  and  6^:3  -  6x^  -  S6x. 

6.  12«V  +  120aV  -  132a^x    and     SaV  -  27aV  + 

7.  7:^;*  -  lOa.'^s  +  3aV  -  4^^^  +  4«4  and  82;'*  -  13aa^ 

8.  25a;*  -i-5x^-x-l  and  20a;4  +  x^  -  1. 

9.  1  -  4a;3  +  3a;4  and  1  +  a;  -  a;^  -  6x^  +  4a;*. 

II. 
Work  the  last  nine  and  also  the  following  examples  by 
synthetic  division: 

10.  11a;*  +  24a;3  +  125  and  x^  +  24a;  +  55. 

11.  2a;5  -  lla;2  -  9  and  4:X^  +  11a;*  +  81. 

12.  a;5  +  lla;3  -  54  and  ^  +  11a;  +  12. 

13.  x^  —  2x^  —  x-\-  2,  x^  —  x^  —  4:X-\-  4,  and  a^  —  7x-\-  6. 

14.  a;*  -  Qx^  +  8a;  -  3,      a;*  -  2a:3  _  7^2  _j_  20a;  -  12, 
and  a;*  —  4a;2  +  12a;  —  9. 

15.  Multiply  3a;"*  -  4a;'"  "  ^  +  5a;"  +  ^  by  6x^  +  7a;"*  +  K 

16.  Multiply  a;"  -  Sx^  +  5a;3  by  4a;*  -  6:^-^ 

EXERCISE  LXIV. 

I. 
Ex.  At  what  time  after  5  o'clock  will  the  minute-hand 
of  the  clock  be  ten  minutes  ahead  of  the  hour-hand  ? 


HIGHEST  COMMON  FACTORS. 


141 


In  examples  about  the  position  of  the  hands  of»a  clock, 
it  is  best  to  draw  a  circle  to  represent  the  clock-dial,  and  to 
mark  on  it  the  positions  of  the  hands  at  the  beginning  of 
the  hour  specified.  Then  note  the  number  of  minute-spaces 
between  the  hands  at  this  time,  and  let  x  denote  the  num- 
ber of  minute-spaces  that  the  minute-hand  must  pass  over 
before  it  comes  into  the  required  position.  Then,  since  the 
minute-hand  goes  12  times  around  the  dial  while  the  hour- 
hand  is  going  once  around  it,  x/1^  will  denote  the  number 
of  minute-spaces  passed  over  by  the  hour-hand  in  the  same 
time. 

Then  x  will  equal  the  number  of  minute-spaces  between 
the  hands  at  the  beginning  of  the  hour  plus  x/1%  minus 
the  number  of  spaces  the  hands  are  required  to  be  apart 
when  the  minute-hand  is  required  to  be  behind  the  hour- 
hand  ;  and  x  will  equal  the  number  of  minute-spaces  be- 
tween the  hands  at  the  beginning  of  the  hour  plus  x/1% 
plus  the  number  of  spaces  the  hands  are  required  to  be 
apart  when  the  minute  hand  is  required  to  be  ahead  of  the 
hour-hand. 

Thus,  in  the  example,  the  minute-hand  will  be  at  XII 
at  the  beginning  of  the  hour  specified, 
and  the  hour-hand  at  V,  and  there 
would  be  25  minute-spaces  between 
them.  While  the  former  is  moving 
over  i\\Q  x  spaces  to  its  required  posi- 
tion of  10  minute-spaces  ahead  of  the 
hour-hand,  the  hour-hand  will  move 
over  x/1%  spaces.     Therefore 

a;  =  25  +  ^/12  4- 10; 
11/12^;  =  35, 
X  =  38^. 

That  is,  the  minute-hand  would  be  in  the  required  po- 
sition at  dSf^  minutes  past  five. 


XII 


142  HIGHEST  COMMON  FACTORS. 

Had  tiie  question  been,  at  what  time  after  5  o'clock  will 
the  minute-hand  of  the  clock  be  ten  minutes  behind  the 
hour-hand,  we  would  have  had 

x  =  25-{-  x/12  -  10; 
.-.  ll/12a:  =  15, 

X  =  Uj\. 

1.  At  what  time  after  3  o'clock  is  the  minute-hand  of 
the  clock  18  minutes  ahead  of  the  hour-hand  ? 

2.  At  what  time  after  7  o'clock  is  the  hour-hand  20 
minutes  behind  the  minute-hand  ? 

3.  At  what  time  after  9  o'clock  is  the  hour-hand  15 
minutes  behind  the  minute-hand  ? 

4.  At  what  time  nearest  to  2  o'clock  is  the  minute- 
hand  15  minutes  behind  the  hour-hand  ? 

5.  At  what  time  between  4  and  5  o'clock  are  the  hour 
and  minute  hands  at  right  angles  ? 

6.  The  sum  of  the  two  digits  of  a  number  is  8,  and  if 
36  be  added  to  the  number  the  digits  will  be  interchanged. 
What  is  the  number  ? 

7.  If  the  first  of  the  two  digits  of  a  number  be  doubled 
it  will  be  3  more  than  the  second,  and  the  number  itself 
is  6  less  than  five  times  the  sum  of  its  digits.  What  is  the 
number  ? 

8.  A  courier  who  goes  at  the  rate  of  40  miles  in  eight 
hours  is  followed  after  10  hours  by  a  second  courier  who 
goes  at  the  rate  of  72  miles  in  9  hours.  In  how  many  hours 
will  the  second  overtake  the  first  ? 

II. 

9.  A  courier  who  goes  at  the  rate  of  31^  miles  in  five 
hours  is  followed,  after  eight  hours,  by  a  second  courier 


HIGHEST  COMMON  FACTORS.  143 

who  goes  at  the  rate  of  22^  miles  in  three  hours.     In  how- 
many  hours  will  the  second  overtake  the  first  ? 

10.  Ten  years  hence  a  boy  will  be  four  times  as  old  as 
he  was  ten  years  ago.     How  old  is  the  boy  ? 

11.  One  man  is  60  years  old,  and  another  man  is  2/3 
as  old.  How  long  since  the  first  man  was  five  times  as  old 
as  the  second  ? 

12.  A  father  is  four  times  as  old  as  his  son,  and  four 
years  ago  the  father  was  six  times  as  old  as  his  son.  What 
is  the  age  of  each  ? 


CHAPTER  XIII. 
LOWEST  COMMON  MULTIPLE. 

109.  Lowest  Common  Multiple. — A  common  multiple 
of  two  or  more  expressions  is  an  expression  which  is  exactly 
divisible  by  each  of  them. 

The  loivest  common  multiple  of  two  or  more  expressions 
is  the  expression  of  the  lowest  dimensions  which  is  exactly 
divisible  by  each  of  them.  The  lowest  common  multiple  is 
usually  denoted  by  the  letters  L.  C.  M. 

110.  To  Find  L.  CM.  by  Inspection.  —  The  lowest 
common  multiple  of  two  or  more  expressions  must  evidently 
contain  every  factor  of  each,  and  each  of  these  factors  the 
greatest  number  of  times  that  it  occurs  in  any  one  of  them, 
otherwise  it  would  not  be  divisible  by  each  expression. 

e.g.  Let  ^a^V^c,  Qa%^G^d,  and  ^h^c^e  be  the  numbers 
whose  L.  C.  M.  is  required.  To  be  divisible  by  each  of 
these  expressions  the  required  expression  must  contain  the 
factors  2,  3,  a,  h,  c,  d,  and  e,  and  it  must  also  contain  the 
first  of  these  once,  the  second  twice,  the  third  four  times, 
the  fourth  four  times,  the  fifth  three  times,  the  sixth  once, 
and  the  seventh  once.     The  L.  C.  M.  is  iMh^c^de, 

Hence  we  have  the  following  rule  for  finding  the  lowest 
common  multiple  of  two  or  more  expressions  which  may  be 
factored  by  inspection : 

Find  all  the  different  factors  of  each  expression,  a7id  take 

■  each  of  these  factors  the  greatest  number  of  times  which  it 

occurs  in  any  of  the  expressions,  or  to  the  highest  degree  that 

it  has  in  any  of  the  expressions,  and  find  the  product  of  these 

factors, 

X44 


LOWEST  COMMON  MULTIPLE.  145 

EXERCISE  LXV. 

Find  the  L.  0.  M.  of  the  following  expressions: 

I. 
1.     lSa%%  U%H\  and  taHK 
•    2.     3a;^?/2^  bxyh^y  16x^yh,  and  20a!^yh\ 

3.  x^  —  y^,  xy  —  y"^,  and  xy  -f-  y"^. 

4.  :z;^  —  2a;  —  15,  x^  —  9,  and  x^  —  ^x -{■  15. 
6.     5a;  H-  35,  x^  -  49,  and  x^  +  14^;  +  49. 

II. 

6.  x^  -X-  20,  a;2  +  3a;  -  40,  and  j?  -\-  V2x  4-  32. 

7.  2x^  -X-  1,  2a;2  -|-  3a;  +  1,  a;^  -  1,  4a;^  -  5a;2+ 1. 

8.  12a;  -  36,  x^  -  9,  x^  -  6x  -\-  6. 

9.  x^  —  dx  +  2,  a;'-^  —  5a;  -|-  6,  and  a;^  —  4a;  +  3. 
10.     x^  —  Qax  -f  9a^,  7?  —  ax  —  6a^,  and  3a;^  —  12^^^. 

111.  To  Find  L.  C.  M.  by  Division. — Since  the  highest 
common  factor  of  two  expressions  contains  every  factor 
common  to  the  expressions,  if  two  expressions  be  each  di- 
vided by  their  highest  common  factor,  the  quotients  ob- 
tained will  contain  no  common  factors.  Hence  the  L.C.M. 
of  the  two  expressions  will  be  the  product  of  these  quotients 
and  their  H.  C.  F. 

e.g.  Find  the  L.  C.  M.  of 

a;3  4-  a;2  -  2  and  x^  +  2a;2  -  3. 
The  H.  C.  F.  of  these  two  expressions  is  a;  —  1. 
(a;3  +  a;2  -  2)  -^  (a;  -  1)  =  a;^  +  2a;  +  2, 
and  (a;3  +  1x^  -  3)  -=-  i^x  -  l)  r=  x^  +  3a;  +  3. 

a;3  +  a;2  -  2  =  (a;  -  \){p?  +  2a;  +  2), 
and  a;3  +  2a;2  -  3  =  (a;  -  l)(a;2  +  3a;  +  3). 


146  LOWEST  COMMON  MULTIPLE. 

Since  x^  -\-  2x  -\-  2  and  x^  -\-  dx  -\-  3  have  no  common 
factor,  {x  -  l){x^  +  2a;  +  2)(a^  +  3a;  +  3)  must  be  the 
L.  0.  M.  otx^  +  x^-2  and  a^  +  2x^  -  3. 

In  general,  let  A  and  B  stand  for  any  two  expressions, 
and  let  h  stand  for  their  H.  0.  F.  and  I  stand  for  their 
L.  0.  M.,  and  let  P  and  Q  be  the  quotients  when  A  and  B 
respectively  are  divided  by  ^;  so  that 

A  =  P.h    and    B=  Q.h. 

Since  h  is  the  H.  C.  F.  of  ^  and  B,  P  and  Q  can  have 
no  common  factors.  Hence  the  L.  0.  M.  of  ^  and  B  must 
hQ  P  X  Qxli,  or 

I  =  PQh; 
or 

^,     on     ,     B 

h  h 

Hence  the  L.  C.  M.  of  two  expressions  may  he  found 
hy  dividmg  either  one  of  the  expressions  by  their  H.  C.  F., 
and  multiplying  the  quotient  by  the  other  expression. 

Also,  since 

T_AX  B 

I  X  h  =  A  X  B. 
That  is,  the  product  of  any  two  expressions  is  equal  to 
the  product  of  their  H.  C.  F,  and  L.  C.  M. 

EXERCISE  LXVI. 

Find  the  L.  0.  M.  of  the. following  expressions: 
I. 

1.  Qx^  —  5ax  —  (ja^  and  4:X^  —  2ax'^  —  Qa^. 

2.  4«2  -  5ab  +  b^  and  3a^  -  da^  +  ab^  -  b^ 

3.  Sx^  -  13x2  +  23x  -  21  and  Qx^  -^  x^  -  Ux  +  21. 

4.  x'  -  lla;2  +  49  and  7x^  -  iOx^  +  7ox^  -  ^Ox  -f  7. 


LOWEST  COMMON  MULTIPLE.  147 

5.  x^  4-  Qx^  -I-  ll.^.  _|_  G  aud  x"-  +  x^  -  4x^  -  4x, 

6.  x^  -  x^  +  8ic  -  8  aud  x^  +  4:6-3  -  Sx^  +  24a;. 

II. 

7.  8^3  -  l8ab^  Sa^  +  Sd^b  -  Qah^,  and  4ft2-8«&  +  3^>l 

8.  x^  -Ix^  12,  3a:2  -  6:?;  -  9,  and  ^x^  -  62;2  -  Sx. 

9.  8a;3  +  27,  16a;*  +  36^2  +  81,  and  Qx^  -  hx  -  6. 

10.  x^  —  Qxy  +  9^'^  x?  —  xy  —  6?/^,  and  3^;^  —  121/2. 

11.  Multiply  x"^  +  a;"  by  x"^  —  x^. 

12.  Multiply  'Sa^'x "  —  ^oTx''  by  3a"a;"*  +  4(x'"iC". 

13.  Divide  a;  - »"  +  ^  by  a:"*  +  ^ 

14.  Divide  4<x'«a;  -  S"*  +  p  by  2a"a;-"»-  ^ 

EXERCISE  LXVII. 
I. 

1.  At  what  time  after  10  o'clock  will  the  minute-hand 
of  a  clock  first  be  20  minute-spaces  ahead  of  the  hour- 
hand  ? 

2.  A  courier  sets  out  from  a  city  and  travels  at  the  rate 
of  8  miles  an  hour,  and  3  hours  later  a  second  courier  sets 
out  from  the  same  city  and  follows  the  first  along  the  same 
road,  travelling  at  the  rate  of  10  miles  an  hour.  In  how 
many  hours  will  the  second  courier  overtake  the  first,  and 
how  far  will  each  have  travelled  ? 

3.  A  courier  sets  out  from  a  city  and  travels  10  miles 
an  hour.  Four  hours  later  a  second  courier  sets  out  from 
the  same  place  and  travels  along  the  same  road  and  over- 


148  LOWEST  COMMON  MULTIPLE. 

takes  the  first  courier  in  20  hours.     How  fast  does  the  sec- 
ond courier  ride,  and  how  far  does  each  go  ? 

4.  Two  bodies,  A  and  B,  are  moving  around  concentric 
circles  in  the  same  direction,  and,  as  seen  from  the  common 
centre  of  the  circles,  they  are  together  every  50  days.  A  is 
on  the  outer  circle,  and  is  longer  in  going  around  than  B, 
which  is  on  the  inner  circle.  A  goes  around  his  circle  in 
20  days.    How  long  does  it  take  B  to  go  around  his  circle  ? 

Let  X  =  number  days  it  takes  B  to  go  around. 

- —  =  number  of  degrees  B  goes  over  in  a  day. 

Also  -^r^  =  number  of  degrees  A  goes  over  in  a  day, 

and -^  =  number  of  degrees  gained  by  B  in 

one  day. 

In  50  days  B  must  evidently  gain  360°  on  A. 

— -  =  number  of  degrees  gained  by  B  in  one  day. 

360  _  360  _  360   ■        1^      J_  _  J_ 
"^  ~  ^"  ~   50  '   ^^  ^  ~  20  ~  50  • 

.  •.      -  =  -|-„  or  Ix  =  100,  and  x  =  14f . 
X       100 

II. 

5.  Suppose,  in  the  last  example,  A  went  around  his 
circle  in  the  shorter  time,  then  in  what  time  would  B  go 
around  ? 

6.  Two  bodies,  A  and  B,  move  around  two  concentric 
circles  in  the  same  direction  and  are  together  every  60  days. 
A  is  on  the  outer  circle  and  B  on  the  inner,  and  A  goes 
around  its  circle  in  40  days.   If  B  moves  over  more  degrees 


LOWEST  COMMON  MULTIPLE.  149 

a  day  than  A  does,  how  long  will  it  take  B  to  go  around  its 
circle  ? 

7.  If  in  the  last  example  A  goes  over  more  degrees  a 
day  than  B  does,  how  long  will  it  take  B  to  go  around  ? 

8.  Divide  x^"^  +  «/^"  by  x"'  +  ?/". 

9.  Divide  x^"^  —  if""  by  x""'  —  «/". 


CHAPTEE  XIV. 
FRACTIONS. 


112.  The  Symbol-. — When  the  operation  of  division 

is  indicated  by  placing  the  dividend  over  the  divisor  with  a 
horizontal  line  between,  the  symbol  is  called  a  fraction,  the 
dividend  being  called  the  numerator  and  the  divisor  the 

denominator.    Thus,  -  is  a  fraction,  a  is  its  numerator  and 

0 

l  is  its  denominator.    The  quotient  which  results  from  the 
division  is  the  value  of  the  fraction.    In  the  type,  7-,  a  and 
1)  stand  for  any  integral  expression,  however  complicated. 
By  definition,  -  =  a  -^  h.     Therefore 


-Xh  =  a-^hxl)  =  a, 
h 

That  is,  the  multiplication  of  a  fraction  by  its  denominator 
produces  its  numerator. 

When  the  numerator  is  a  polynomial,  the  horizontal 
line  or  bar  of  the  fraction  must  be  considered  as  a  sign  of 
aggregation,  showing  that  the  numerator  as  a  whole  is  to 
be  divided  by  the  denominator. 

In  the  various  operations  on  fractions  we  assume  that 
the  associative,  distributive,  and  commutative  laws  which 
have  been  demonstrated  for  integers  apply  also  to  the  sym- 

150 


FBAGTI0N8.  151 

bol  T-.  Having  made  this  assumption,  we  proceed  to  en- 
quire what  addition,  multiplication,  and  other  operations 
on  fractions  mean  if  they  obey  the  same  laws  as  the  corre- 
sponding operations  on  numbers. 

113.  Theorem  I.     The  denominator  of  a  fraction  is 
distributive  among  the  terms  of  its  numerator. 

-.,  .           .,,               a-^b      a       b 
It  IS  required  to  prove  =  -  -| — . 

c  c       c 

By  definition  x  c  =  a  -\-b. 

By  the  distributive  law 

(a       b\        a  ,   b  ,   , 

\c       c  J        c  c 


-b 


c>'-\-b  ^^     _{a    I   b\ 


a  -\-  b  _a       b 

c      ~  c       c' 

It  is  thus  seen  that  this  theorem  is  a  consequence  of  the 
assumption  that  the  distributive  law  of  multiplication  holds 
for  fractional  symbols. 

Hence  the  denominator  of  a  fraction  is  distributive 
throughout  the  terms  of  the  numerator. 

And,  conversely,  the  algebraic  sum  of  any  number  of 
fractions  with  the  same  denominator  is  the  fraction  whose 
numerator  is  the  algebraic  sum  of  the  numerators  of 
several  fractions,  and  whose  denominator  is  their  common 
denominator. 

The  sign  before  a  fraction  may  always  be  regarded  as 
belonging  to  the  numerator  as  a  whole,  and  it  must  be  so 
regarded  in  finding  the  algebraic  sum  of  the  numerators 
of  fractions  which  have  the  same   denominator.     Thus, 


152  FRACTIONS. 


+  r  =  —f—y  and  —  ^  =  —r--     The  value  of  a  fraction  is 

0  0  0  0 

to  be  regarded  as  a  quotient,  and  when  the  divisor  is 
positive  the  sign  of  the  quotient  is  the  same  as  that  of  the 
dividend.      Hence,    if  a  and   h  both   represent   positive 

quantities,  —  -  =  — —  =  — -,  but  is  not  =  — -.    That  is, 

0  0—0  —  0 

the  minus  sign  before  a  fraction  may  be  regarded  as  be- 
longing to  either  the  numerator  or  denominator  as  a  whole, 
but  not  to  both. 

The  same  is  evidently  true  when  both  a  and  h  repre- 
sent negative  quantities,  or  when  one  of  them  represents  a 
negative  quantity  and  the  other  a  positive  quantity. 

For  —  — ,    — zT-j    and  — —  each  evidently  represent 

the  same  negative  quantity;  and  —  :p7,  ,  and  — jpr 

each  evidently  represent  the  same  positive  quantity,  as  do 
also 7,  — 7-  ,  and 


-  -b. 
To  illustrate  by  numerals  : 

-  ^-  -  2,    =^=  -  2,    and  -1^  =  -  2; 

_:^8_2,    -:^:=2,    and   ^=2; 

It  must  be  borne  in  mind  carefully  that,  in  finding  the 
algebraic  sum  of  the  numerators  of  fractions  which  have 
the  same  denominator,  all  the  signs  of  the  numerator  of 
every  fraction  which  has  a  minus  sign  must  be  changed. 


FRACTIONS.  153 

EXERCISE    LXVIII. 
I. 

1.  Write ^ as  the  sum  of  three  separate 

fractions. 

2.  VVrite ^— r— 7 ' —    as    the    sum    of   four 

a  -\-  b 

fractions. 

3.  Write  - — — ^  —  - — —7  +  -; — — Y  as  one  fraction. 

'^a  -{-b       2a-{-  b       2a  -\-  b 

,-,^  .,    3.T  +  5      4:^;  +  6      6x  —  a  ,    7x  —  c 

4.  Write  — -r r a :: as  one 

4c  4c  46*  4c 

fraction. 

II. 
6.     Write 

2a -\- 3b -c      5a -7b  ^11      da -{- 5b  -  7      lla  -  d 
x^-3      ^        x^-3  x^-S  x^-3 

as  one  fraction. 

6.     Write 

3x  -  4t(a  -\-b)       5x^  7{a  ■\-b)        7x  -  5(a-  c) 
a2  -J)i  a^-  ])i  a"  -  b^ 

as  one  fraction. 

114.  Theorem  II.  The  value  of  a  fraction  is  not 
altered  by  multiplying  its  numerator  and  denominator  by 
the  same  quantity. 

It  IS  required  to  prove      T~~i- 

By  the  commutative  law  t  •  mb  —  -r- .  bxm  =  am  =  ma. 

0  0 


154  FRACTIONS. 

By  definition 


ma 
mb 

.  mb  ■■ 

=  ma 

a 

1' 

mb  — 

ma 

mb' 

mb. 

a 

ma 

I  " 

mb' 

115.  Theorem  III.  The  value  of  a  fraction  is  not 
altered  by  dividing  its  numerator  and  denominator  by  the 
sa7ne  quantity. 

T.  '  .    /,  X  a  ^  m     a 

It  IS  required  to  prove  -^ =  7. 

^  ^         b  -r-  m      b 

-o    XT,    1    X  XI,  a^m       (a-^  771)771 

By  the  last  theorem =  -77 { — . 

•^  b  -^  m       {b  -^  7n)m 

^   ,  -      ,  ^   ., .  (a  -=-  m)m       a 

But  by  definition  77 ^-.  —  ^. 

-^  {b  -i-  m)7n       b 

116.  It  follows  from  Theorem  III  that  a  fraction  may 
be  simplified  without  altering  its  value  by  the  rejection  of 
any  common  factor  from  its  numerator  and  denominator. 

Thus  the  fraction  -73-  takes  the  simpler  form  —7-3,  when 

the  factor  Xy  which  is  common  to  its  numerator  and  denom- 
inator, is  rejected. 

A  fraction  is  said  to  be  in  its  loivest  terms  when  its 
numerator  and  denominator  have  no  common  factors. 

A  fraction  may  be  reduced  to  its  lowest  terms  by  re- 
moving, or  cancelling,  the  common  factors  one  after  an- 
other from  the  numerator  and  denominator  by  inspection, 
or  by  dividing  the  numerator  and  denominator  by  their 
H.  C.  F. 

When  the  numerator  and  denominator  of  a  fraction 
are  polynomials  which  can  be  factored  by  inspection,  it  is 


FRACTIONS.  155 

best  to  write  them  as  factored,  and  then  to  cancel  their 
common  factors. 

3a;^  +  a;  -  2  _  (3a;  -  2)(a;  +  1)  _  82;  -  2 
®*^'     '■Zx^  -x-d~  {2x  -  d)(x  +  1)  ~  2:r  -  3  • 

It  is  not  worth  while  to  divide  the  numerator  and  de- 
nominator by  their  H.  C.  F.  except  in  cases  where  their 
common  factors  cannot  be  discovered  by  inspection. 

EXERCISE  LXIX. 

Reduce  the  following  fractions  to  their  lowest  terms. 


12A  UaWc 


X  —  a 


4. 


x^  —  a' 


ax  —  a^  '     x^  -\-  ax 

3x^  -  9x^y  Saf^x^  -  16a^x^ 


7. 


7x^  -  21xY  ^ali^x^  -  16Z»V ' 

x^  -\-x-'2Q  x^  -  36 

x''  -\\x^  28*         ®-    ^:i"3^~irT8- 


II. 

4a7^  —  16  ^x^  —  7.T 

10. 


%x^  -  2x  -  12' 
x^  +  3x  -  28 


11.        2    ■    o ^-  12. 


2x?  +  X 

-6  ' 

2:3  +  27 

X^~^' 

4.x?  -  ^x 

+  3 

.     Qx^  -\-  xy  —  y^ 
^^'     8^2  _^  2x1/  -'^"  ^*-     4:7-2  _^  4^  _  3  • 

117.  Reduction  of  Fractions  to  a  Common  Denominator. 

— Two  or  more  fractions  may  be  reduced  to  equivalent 
fractions  with  a  common  denominator  by  finding  the  L.  C. 


156  FRACTIONS. 

M.  of  the  denominators  for  the  common  denominator,  and 
dividing  this  by  each  of  the  old  denominators  in  turn,  and 
multiplying  each  numerator  by  the  corresponding  quotient 
for  the  numerator. 

N.B. — This  is  equivalent  to  multiplying  the  numerator 
and  denominator  of  each  fraction  by  the  quotient  obtained 
by  dividing  the  L.  C.  M.  of  all  the  denominators  by  its  own 
denominator;  and  hence  the  value  of  the  fractions  will 
not  be  altered.     (Why  ?) 

An  integer  may  be  regarded  as  a  fraction  whose  denom- 
inator is  one.  Hence  an  integral  term  may  be  reduced  to  a 
fraction  with  any  denominator  by  multiplying  it  by  the  re- 
quired denominator  and  placing  the  product  obtained  over 
the  denominator. 

Of  course  any  fraction  may  be  reduced  to  an  equivalent 
fraction  with  any  required  denominator  (which  is  a  mul- 
tiple of  its  own)  by  multiplying  the  denominator  of  the 
fraction  by  the  factor  which  will  produce  the  required  de- 
nominator, and  the  numerator  by  the  same  factor.  Such  a 
factor  may  be  obtained  by  dividing  the  required  denomina- 
tor by  the  old  one,  or,  often,  by  simple  inspection. 

EXERCISE  LXX. 
I. 

1.  Reduce  -^  to  an  equivalent  fraction  whose  denom- 
inator is  9«c^. 

2.  Reduce  — to  an  equivalent  fraction  whose 

denominator  is  'iWu^. 

X  —  S 

3.  Reduce  to  an  equivalent  fraction  whose  de- 

X  -J-   i 

nominator  is  2;^  -|-  :^  —  42. 


FRACTIONS.  157 


4.     Reduce  ~ ~  to  an  equivalent  fraction  whose  cle- 

nominator  is  I'^x^  -\-  x  —  Q. 

5^ ij" 

6.     Reduce  ^r to  an   equivalent  fraction   whose 

ZX  —    D 

denominator  is  %x^  —  34a;  +  30. 

6,     Reduce   ^a^x  to  an  equivalent  fraction  whose  de- 
nominator is  ba^x^. 


II. 

7.  Reduce  2^V  to  an  equivalent  fraction  whose  de- 
nominator is  3  —  laW^. 

8.  Reduce  3:?;  —  5  to  an  equivalent  fraction  whose  de- 
nominator is  7r?;  +  8. 

9.  Reduce  5a;  —  7  to  an  equivalent  fraction  whose  de- 
nominator is  6  —  Zx. 

10.     Reduce  3a;  +  8  to  an  equivalent  fraction  whose  de- 
nominator is  9  —  hx. 


5  Ix 

11.  Reduce  ^r— 7^  and  —3-   to  equivalent  fractions  with 

a  common  denominator. 

12.  Reduce  and to  equivalent  fractions 

X  ~j~    TC  X   4: 

with  a  common  denominator,  and  find  their  sum. 

Reduce  the  following  terms  to  equivalent  fractions  with 
a  common  denominator,  and  then  the  whole  to  a  single 
fraction : 

,    ,    a;+  7      a;- 8 

13.  1  +  — ~- r-T- 


158  FRACTIONS. 

6x  +  6 


14.     ^r- 


15.     dx 


2a  4:a^ 

2x-3       4a:  -  6 


3x-\-4:       5x-2 


16     Reduce  —  ^j-^  +  -  to  a  single  negative  fraction. 
4fl2  '  Of 

II. 

257,2       >yQ 

17.  Reduce  -  ^tts  +  ^  ^0  a  single  negative  fraction. 

Sba'*       oa 

18.  Reduce  1 ■ — j-r to  a  single  positive  frac- 
tion. 

19.  Divide  x^""  —  x^"-  by  x"^  +  x"". 

20.  Divide  x^"^  +  x^""  by  x"'  +  a;". 

118.  Theoeem  III.     The  product  of  two  fractions  is 

the  product  of  their  7iumerators  divided  hy  the  product  of 

their  denominators. 

^    .  .     -,  ,  a       c      ac 

It  IS  required  to  prove  t'^-3  —  -ti- 

By  the  commutative  law  -X-j.hd  =  -.h  X  -^.  d  =  ac. 

•^  h      d  b  d 

ac 
By  definition  -:j—,.bd=ac. 

^  bd 

b      d  bd 

a       c  _  ac 
b      d~'bd' 

Hence  the  product  of  two  fractions  is  another  fraction 
whose  numerator  is  the  product  of  their  numerators,  and 
whose  denominator  is  the  product  of  their  denominators. 


FBAGTI0N8.  159 

The  product  of  any  number  of  fractions  may  be  found 
by  first  finding  the  product  of  any  two  of  them,  and  then 
of  the  resulting  fraction  and  a  third,  and  so  on  to  the  end. 
The  resulting  product  evidently  will  be  the  fraction  whose 
numerator  is  the  product  of  the  numerators  of  all  the 
given  fractions  and  whose  denominator  is  the  product  of 
their  denominators.     Thus, 


b      d      f      h      bd     f      h     bdf     h     bdfU 
Hence 

-        \b)=b''b=¥'   ^^^   \b)  ^¥' 

Cor.  1.    A  fraction  may  be  multiplied  by  a  quantity 
by  multiplying  its  numerator  by  that  quantity. 

For  let  7-  be  a  fraction  and  c  be  the  quantity  by  which 
b 

it  is  to  be  multiplied,     c  may  be  written  as  the  fraction  -. 

a  a      c      ac 

b  bib 


Also,  by  the  Commutative  Law,  7-  X  c  =  c  X  7-. 

0  0 


a      ac 

Cor.  2.    A  fraction  may  be  multiplied  by  a  quantity 
by  dividing  its  denominator  by  that  quantity. 

For  let  -  be  a  fraction,  and  c  be  the  quantity  by  which 

it  is  to  be  multiplied. 


160  FRACTIONS. 


Then  -  x  c  =  j-.      Multiplying  both  the  numerator 

and  denominator  by  -,  we  have 

1 
ac .  — 

c        a  a 

or 


,     1  b  '     ^^     i^c 

0  .—  — 

c  c 

EXERCISE  LXXI. 


N.B. — In  multiplying  fractions  by  integers  or  fractions 
it  is  best  to  cancel  common  terms  as  in  arithmetic. 
Find  the  following  products  ; 


dz       4:X^y^       z ' 
c?  —  x^  a^x  -\-  ax^  2{a  —  x) 


2ax  (T  —  2ax  +  x;^         c?  -\-  ax 

a^  4-  ax  a^  —  a^ 

3.     -^ — 5-X 


a!'^  —  x^      ax{c?  -\-  ax-\-  x^)' 

c^  —  x^       c?  —  if-       I  ax   \ 

4.       1 X  r^X^H . 

a-\-  y        ax  -\-  x^      \     '  a  —  xJ 

II. 

ax  —  x^  «2  _|_  ff^x 


a^  —  2ax  -{-x^      a^  +  2ax  +  x^' 


FBACTI0N8.  161 


•■  e+^^)(:-+«--)' 


10. 

42)2  -  16a;  +  15  ^     Q?  -6a; -7     ^         4a;2 


2a;2  +  3a;  +  l        2a;2  -  17a;  +  21       4a;2  -  20a;  +  25* 

119.  Reciprocals. — The  reciprocal  of  a  fraction  is  the 

c  .    d 
fraction  inverted.     Thus,  the  reciprocal  of  -^  is  -. 

(t     c 

120.  Theorem  IV.  To  divide  one  fraction  hy  another 
is  equivalent  to  multiplying  the  first  fraction  hy  the  recip- 
rocal of  the  second. 

It  is  required  to  prove  that  -  -^  -  =  -  x  — . 
^  ^  0      d      b      c 

By  definition  of  division,      -  ^  -  x  -  =  7-. 
^  1)       d      d      h 

By  Theorem  III  and  the  associative  law  of  multiplica- 
tion, 

a      f?      c  _ «    cd  _a 
h      G      d      b'  cd      b' 


(a  ^c\c  _fa      d\c 

[b   '  d)d~[b^c)d' 
a  _^G  ^  a      d 
b    '  d      b      c  ' 

Hence  to  divide  one  fraction  by  another,  we  invert  the 
divisor,  and  then  proceed  as  in  multiplication. 

Cor.  1.  A  fractio7i  may  be  divided  by  a  quantity  by 
multiplying  its  denominator  by  the  quantity. 

For,  let  T-  be  a  fraction  and  c  be  the  given  quantity. 
Then  will  ?  ^  c  = 


h  b  xc 


162  FRACTIONS. 


c,.  c  ,         a  a       1       a 

Since  c  =  -,  we  nave  —  -hc  =  -X-=7-. 
1  0  0       0       be 

CoE.  2.  A  fraction  may  he  divided  hij  a  quantity  hy 
dividhig  its  numerator  hy  the  quantity. 

For,  let  T-  be  a  fraction  and  c  be  the  given  quantity. 
Then  -  -^  c  —  -r-,  and  multiplying  the  numerator  and  de- 
nominator by  -,  we  have 

1  a 

a .  -  - 

a  c  c  a  -7-  c 

-j_  c  = =:  — -     or      -— . 

h  .,      1  h  h 

he .  - 
c 

Cor.  3.  To  divide  a  quantity  hy  a  fraction  we  multi- 
ply the  quantity  hy  the  reciprocal  of  the  fraction. 

Let  fl^  be  a  quantity,  and  ^  be  a  fraction.     Then  will 


«^^  =  « 

xf 

a 

c        a       c 

a       d       ad              d 
-  X  -  =  —  =  aX  -0 
Ice                c 

EXERCISE 

LXXII. 

Perform  the  operations  indicated  in  the  following  ex> 

amples : 

* 
I, 

Ux'^  -  7x 

2x-\ 

*•     12^:3  +  24:^2 

'  x^  +  %x 

a^^  +  dab 

^  ah^Z 

r-  r : —m 

4a^  -  X      •  2a  -1-  1* 


3. 


FRACTI0N8. 

«2- 

-121 

«  +  11 

a' 

-4 

•     «  +  2  * 

2a^  +  13x 

+  15   . 

2^2  _^  11^.  _^  5 

4:^2  _ 

9 

4a;^  -  1       • 

x'- 

-  14:«  - 

-  15 

x^  -  12a;  -  45 

x^ 

-^x- 

-45    • 

x^  -  ex  -  27  * 

(10 

+  lla; 

-  Qx^) 

9a;2-4 
'    4 -3a;* 

(15. 

,;2_  19a; +  (3) 

18  -  18a;  -  20a;2 

2a;  +  7 

{x'^ 

-%x- 

-63)  -^ 

a;2  +  2a;  -  35 

163 


8. 


121.  To  Multiply  Several  Fractions  by  a  Factor  which 
will  Cancel  all  their  Denominators. — If  each  of  several 
fractions  be  multiplied  by  the  L.  C.  M.  of  their  denomina- 
tors, there  will  be  introduced  into  the  numerator  of  each 
fraction  a  factor  which  will  cancel  its  denominator,  and  the 
resulting  products  will  be  the  product  of  the  numerator  of 
each  fraction  and  all  the  factors  of  the  L.  C.  M.  of  the  de- 
nominators except  the  denominator  of  the  fraction.  We 
may  therefore  obtain  these  products  by  dividing  the  L.  C. 
M.  of  the  denominators  by  each  denominator  and  multiply- 
ing the  numerator  of  each  fraction  by  the  resulting  quo- 
tient. 

e.g.  Find  the  product  which  would  result  from  multi- 
plying each  of  the  following  fractions  by  the  L.C.M.D. : 

a;  +  7  a;-8  ^  a;  +  9 

and 


Q?-\-'^x-  10'     a;2  -  8a;  +  12  x^  -  x  -  30' 

Factoring  the  denominators,  we  get 
x-\-l  a;—  8  a;+9 


(x  -  2)(a;  +  5)'         {x  -  2){x  -  6)'  (x  +  5)(a;  -  6)' 


164  FRACTIONS. 

Hence  the  L.  0.  M.  of  the  denominator  is 

{x  -  %){x  +  b){x  -  6). 

Multiplying  each  fraction  by  this  L.  C.  M.,  and  cancel- 
ling the  common  factors,  we  obtain 

{x^l){x-^){x-^6){x-  6) 
\x-^)\x^6) 

(g;- 8)(a;- 2)(a;+  b){x  -  6) 
{x-%){x-Q) 

and  (a;+9)(:.-2)(a;+5)(:.-6) 

{x+b){x-Q) 

or    x^-^x-  42,     x^-'^x-  40,     and    x'^-\-'ix-  18. 

EXERCISE  LXXIII. 

Find  the  products  obtained  by  multiplying  each  frac- 
tion of  the  following  sets  by  the  L.  C.  M.  of  the  denomina- 
tors: 


a;-4 


a?-\-x-6Q'    x^.-\- 11x^^4:'    x^-4:x-21' 
dx-7  5a;  -  4  a;  +  11 


10a;2  -  43a;  +  28 '    16x^  +  8a;  -  16'     6x^  -  13a;  -  28* 

5a;  —  8  6  —  7a;  3  —  a; 

66a;  -  ISa;^  _  q^>  24:X^  -  90a;  +  54'  40a;2  _  86a;  +  42* 

11. 

x-S  x-^  7 


*•      0^-64:'    (^'i-4:X-32'    a^ -\- 4:x -\- 16' 

x-\-7  a;4-6        15 

^'     ie8  +  216'    a;2-36'    3a;2  -  108' 

6.     Divide  x^""  ~  x>  by  x^  —  re*. 


FRACTIONS.  165 

EXERCISE  LXXIV. 

I. 

1.  A  is  four  times  as  old  as  B  and  6  years  ago  he  was 
seven  times  as  old.     What  is  the  age  of  each  ? 

2.  At  what  time  after  3  o'clock  are  the  hands  of  a 
watch  opposite  each  other  for  the  first  time  ? 

3.  Divide  45  into  two  parts  such  that  one  of  them  shall 
be  four  times  as  much  above  20  as  the  other  is  below  19. 

4.  A  man  had  $13.55  in  dollars,  dimes,  and  cents.  He 
had  1/7  as  many  cents  as  dimes,  and  twice  as  many  dollars 
as  cents.     How  many  of  each  kind  had  he  ? 

6.  Divide  313  into  two  such  parts  that  one  divided  by 
the  other  may  give  2  as  a  quotient  and  19  as  a  remainder. 

II. 

6.  A  is  m  times  as  old  as  B,  and  in  c  years  he  will  be 
n  times  as  old.     What  is  the  age  of  each  ? 

7.  At  what  rate  of  simple  interest  will  a  dollars 
amount  to  h  dollars  in  c  years  ? 

8.  The  denominator  of  a  fraction  is  equal  to  four  times 
the  numerator,  diminished  by  41,  and  if  the  numerator  be 
diminished  by  6  and  the  denominator  be  increased  by  9, 
the  value  of  the  fraction  will  be  5/12.  What  is  the  frac- 
tion ? 

9.  At  what  time  after  5  o'clock  are  the  hands  of  a 
watch  together  for  the  first  time  ? 

10.  Divide  n  into  two  parts  such  that  one  divided  by 
the  other  will  give  g'  as  a  quotient  and  r  as  a  remainder. 


CHAPTER  XV. 
CLEARING  EQUATIONS  OF  FRACTIONS. 

122.  Three  Classes  of  Equations  Involving  Fractions. — 

As  we  have  seen,  an  equation  may  be  cleared  of  fractions 
by  multiplying  both  members  by  the  least  common  multi- 
ple of  the  denominators  of  all  the  fractions  in  the  equa- 
tion. 

Equations  involving  fractions  may  be  divided  into  three 
classes : 

1°.  Those  m  loliicli  we  should  clear  of  fractions  at  once 
or  after  maMng  some  slight  reductions. 

2°.  Those  ivhich  might  he  cleared  of  fractions  partially 
and  then  simplified. 

3°.  Those  in  which  some  or  all  of  the  fractions  had 
hetter  he  reduced  to  a  mixed  form. 

Case  1°. 

In  clearing  equations  of  fractions,  it  must  be  borne  in 
mind  that  every  term  of  both  members,  integral  as  well  as 
fractional,  must  be  multiplied  by  the  L.  C.  M.  D. 

In  clearing  equations  of  fractions,  it  is  best  to  express 
the  L.  C.  M.  of  the  denominators  as  factors,  and  also  to 
indicate  the  work  of  multiplication  before  actually  perform- 
ing it.  In  this  way  like  factors  in  tlie  numerators  and 
denominators  may  be  cancelled,  and  the  work  much 
shortened. 

e.g.     Solve   — -— -—  -\ — -  =  0. 

166 


CLEARING  EQUATIONS  OF  FMACTlONS.        167 

L.  0.  M.  D.  (x  +  l){x  +  2){x  +  4). 

{x -j- l){x -^  2){x  +  A)  _  2(x  +  l)(x+2)(x  +  ^) 
x+l  x-^2 

{x  +  l){x-^2)(x  +  ^)_^ 

'  X  -\-   4: 

...  (x-i-2)(x  +  4)-2{x-\-l){x-\-4)-\-{x-^l){x-\-2)  =  0, 
or  a;2  _^  62;  +  8  -  2a;2  _  lOa;  -  8  +  ^c^  +  3a:  +  2  =  0. 

-x-\-2  =  0. 
x  =  2. 

When  all  of  the  fractions  are  written  as  decimals,  it  is 
best  first  of  all  to  reduce  these  to  the  form  of  vulgar  frac- 
tions. 

e.g.  Solve     •^^"•^^'^  -  (.03  -  .02x)  =  .03. 
Reducing  the  decimals  to  vulgar  fractions,  we  have 

_  /J 2^\  _  _3^ 

\100       100/  ~  100' 


^'      IO-ro-100  +  -100=^100-     L.C.M.D.  =  100. 

5  X  100   a;  X  100   3  X  100  2x  X  100_  3  X  100 
10       10       100  "*"   100   ~   100  ' 

or       50  -  10a;  -  3  +  2a:  =  3. 

-8a:r=-44; 
X  =  5i, 


5 

X 

100 

100 

1 

10 

168         GLEAMtNQ  EQUATIONS  OF  FRACTIONS. 


EXERCISE  LXXV. 

I. 


1.  \x-m^^[^x-i)-l{x-\-i)-ii. 

2.  |(2^-7)-|(^-8)=?^^  +  4. 

3.  ,^(4^  +  1)  -  ^(217  -  ^)  =  45  -  IZi^. 

4.  .03a;  +  .02  =  .0%x  -  .06. 

5.  .Ql{x  -  10)  +  .542:  =  .2(.l  -  .Ix)  -  3(.05  -  .02). 

^  —  X       6  —  X       ^  x^  —  2 

6.  :; i^ =  1  — 


1-x       7-x  7-8a;  +  a;2 

3  1  ^  +  10  _ 


2a;  -  4       x-{-2   '   2x^  —  S 

2  ^       -  1  4a;2  -  1 


**•     1  -2x      2x-7  4:x^-  IQx  +  7' 

II. 

9.     (1  -  22;)(.01  -  .03a;)  -  .23 

=  (.6a;  +  .l)(.la;  -  .1)  -  .03a;. 

.Ola;       X       .Ola;   ,   ^  „, 
''•     -^-30  =  ^^^'^^' 

.03a;  -  .01        .02(a;  -  1)        .Ola;  -  .03   ,    .21 
^^'  .02  .03  .4  ^  .2 

12.  (.la;+.2)2  +  .7(.3a;-.l) 

=  .06(2a;  +  4)  +  (.la;  -  .2)2-. 65. 

4-a;       6-a;       ^  2a;2  +  8 

13.  H o =  2  - 


2  -  a;       8  -  a;  16  -  10a;  +  a;^' 

5 2 11a;      ^ 

^^'     3a;  -  9       a;  +  3  ~^  3a;2  -  27 


CLEARING  EQUATIONS  OF  FRACTIONS.        169 


_  __3 3^  _  ^x{^x  -  17) 

^*-  l-'dx       dx-7  ~  ^x^  -  24a;  +  7 


Case  2°. 

123.  When  the  L .  C .  M.  of  the  denominators  of  all 
the  fractions  which  occur  in  the  equations  is  inconveniently 
large,  it  is  easier  to  multiply  both  members  by  the  L.O.M. 
of  two  or  more  of  the  denominators,  and  then  reduce  as 
much  as  possible  before  proceeding  farther. 

e.g.     Solve  ^-^-^,_^^^^^^^-^+l. 

Multiplying  by  2 (a;  —  1),  we  get 

2^  __  3(2_+|!)  ^  3  _  22;  +  2a;  -  2,        • 
a;  —  1 


or 


2.  ^(^+f)  =  i. 

a;  —  1 

.-. 

2a;(a;  -  1)  -  2(2  +  a;^)  =  a;  -  1. 

•.     2a;2  _  2a;  -  4  -  2x'^  =  x  -  1. 

.-.     3ai=-3. 

.-.     a;=-l. 

EXERCISE  LXXVI. 

Solve  the  following  equations: 

I. 

I  +  - 

;  -  2a;      a;  -  3       1 
10             22     ~  5* 

2. 

4a;  4- 
9 

3      8a;  +  19       7a;  -  29  __ 

18        '   5a;  -  12  ■"    * 

Z 

h^'^ 

3a; +  10       X 
^^"^^      10a; -50 -5' 

170         GLEAMING  EQUATIONS  OF  FEACTI0N8. 

II. 

8a:  +  5      3  -  7a;       IQx  -\-  15  _  2i 
*•     "~Ii  6a; +  2  28        ~  7 ' 

6a;-  7i       1  +  16a;  _  53  -  24a;  _  12g  -  8a; 
^'     13  -  2a;  "^         24       ""12  3       ' 

Case  3°. 

124.  When  the  degree  of  the  numerator  of  any  of  the 
fractions  equals  or  exceeds  that  of  the  denominator,  it  is 
best  in  most  cases  to  write  the  fraction  in  the  mixed  form 
obtained  by  dividing  the  numerator  by  the  denominator 
and  writing  the  remainder  in  the  form  of  a  fraction  after 
the  integral  quotient;   thus: 

x-l       ^  2 


a;+l  a;  +  l' 

5a; +  4  ^  2 


X  —  'S  X  —  S' 

After  writing  the  fractions  as  mixed  numbers,  the  equa- 
tion may  generally  be  considerably  reduced  before  finally 
clearing  of  fractions. 

,     ^,  ,  ,     .^  +  9,        3a;2  +  6 

e.g.   1.   Solyea:  +  3~^-_,^^=3^-^. 

Writing  the  second  fraction  in  the  mixed  form,  we  have 

a;  +  9  ,     a;+6 


3(a;  -  1)  '    3a;  -  1 

0;  +  9  a;  +  6 


•  •     3(a;  ■ 

-1)" 

■  3a;  -  1* 

(a;  +  9)(3a;- 

-1)  = 

:  3(a;  +  6)(a; 

-1). 

3a;2  +  26a; 

—  9  = 

3(a;2  -f  5a;  - 

-6). 

,*, 

11a;  = 

:  -9. 

• 

.     a;  = 

9 
11* 

or 


CLEA1UN6  EQUATIONS  OF  FRACTIONS.        171 

„^,      X  —  1       X  —  2       X  ~  4:       X  —  5 

2.  Solve ^  — = -. 

X  —  Z       X  —  3       X  —  5       X  —  b 

Writing  each  fraction  as  a  mixed  quantity,  we  have 

l  +  -^-(l+^  =  l  +  ^--fl  +  -^). 
■x-2      \^x-SJ  ^x-6       \^x-Qj 

1  1 1     _      1 

X  —  2      X  —  3  ~  X  —  6      X  —  6' 

We  may  now  write  each  member  as  one  fraction  and  get 

X  -3  -x-{-2  _x-Q-x-^5 
{x  -  2)(x  -  3)  ~  Jx'-  6){x  -  6)' 

-  1  _  -1 

{x  -  2)(x  -  3)~  (x-  5){x  -  6)* 

.-.     (x  -  2){x  -  3)  =  (x  -  6){x  -  6). 

.-.     x^  -  6x  +  0  =  a;2  -  II2;  +  30. 

.-.     6a;  =  24. 

.  %       X  =  4:. 

EXERCISE    LXXVil. 

Solve  the  following  equations : 

I. 

x  —  1      x  —  2  a:  +  ^,^  —  '^..o 

^\    ^ZT^-x-b'  ^'     x-3'^x-Q 

x-\-3x-4._  X  3x    _     ^ 

3a; +5   ,   2a; +  4  _ 
^-     3a;  -  5  +  a;  -  2         '^' 

2x  5  2a;  -  5  __ 

®-     2it;  +  1  "^  2a;  -  1  +  2a;  -f  1 


172         CLEABINQ  EQUATIONS  OF  FRACTIONS. 


7. 


10. 


11. 


12. 


II. 

X  — 

1 

2   . 

x-^ 

x-3 

X-4: 

x-4: 
X  -  6' 

X  — 

x-^~ 

x^ 
x-Y 

1^  +  6 

x-\-2 
x-\-3 

x  +  5 
^  x  +  Q 

8- 

hx 

4.x +  3 

=  lh 

2a;- 

-  1 

'    x  +  3 

^  + 

« 

x-\-b 

X  — 

« 

'  X  —  V 

X 

X  -\-  a  — 

b 

a(a  —  h) 

X  — 

a 

x-h 

~  {x-  c){x  - 

d)- 

X  — 

la 

X  —  a 

5a  _  X  —  a 

X  —  9a      X  —  3a      x  —  2a      x  -{-  'Za 
EXERCISE  LXXVIII, 


1.  A  vessel  can  be  emptied  by  three  taps :  by  the  first 
alone  in  3  hours  and  40  minutes,  by  the  second  alone 
in  2  hours  and  45  minutes,  and  by  the  third  alone  in  2 
hours  and  12  minutes.  In  what  time  would  it  be  emptied 
were  it  full  and  all  three  taps  were  opened  together  ? 

2.  A  cistern  can  be  filled  in  15  minutes  by  two  pipes, 
A  and  B,  together.  After  A  has  been  opened  for  5  minutes 
B  is  also  turned  on,  and  the  cistern  is  filled  in  13  minutes 
more.  In  what  time  would  it  be  filled  by  each  pipe  sep- 
arately ? 

3.  A  man  invests  one  third  of  his  money  in  3-per-cent 
bonds,  two  fifths  of  it  in  4-per-cent  bonds,  and  the  remain- 
der of  it  in  5-per-cent  bonds.  His  income  from  his  invest- 
ment is  1180  dpllars.     How  much  had  he  invested  ? 


CLEARING  EQUATIONS  OF  FRACTIONS.        173 

4.  A  man  invested  one  quarter  of  his  money  in  3  - 
per-cent  bonds,  two  sevenths  of  it  in  4-per-cent  bonds,  and 
the  remainder  of  it  in  4^-per-cent  bonds.  His  income  from 
his  investment  was  3450  dollars.  How  much  had  he  in- 
vested ? 

5.  Two  men,  A  and  B,  66  miles  apart,  set  out,  B  45 
minutes  after  A,  and  travel  towards  each  other,  A  at  the 
rate  of  4  miles  an  hour  and  B  at  the  rate  of  3  miles  an 
hour.     How  far  will  each  have  travelled  when  they  meet  ? 

6.  The  second  figure  of  a  number  composed  of  three 
figures  exceeds  the  third  by  5,  and  the  first  digit  is  one 
fourth  of  the  second.  If  the  number  increased  by  3  be 
divided  by  the  sum  of  its  digits,  the  quotient  will  be  22. 
What  is  the  number  ? 

7.  A  number  is  composed  of  three  digits.  The  second 
digit  is  one  half  of  the  third  and  2  smaller  than  the  first. 
If  the  number  be  diminished  by  18  and  then  divided  by 
the  sum  of  its  digits,  the  quotient  will  be  37.  What  is  the 
number  ? 

8.  A  banker  has  two  kinds  of  money.  It  takes  a 
pieces  of  the  first  to  make  a  dollar  and  b  pieces  of  the  second 
to  make  a  dollar.  He  was  offered  d  dollars  for  c  pieces. 
How  many  of  each  kind  would  he  give  ? 

9.  A  and  B  start  in  business  at  the  same  time,  A 
putting  in  3/2  as  much  capital  as  B.  The  first  year 
A  gains  150  dollars  and  B  loses  1/4  of  his  money.  The 
next  year  A  loses  1/4  of  his  money  and  B  gains  300 
dollars;,  and  they  now  have  equal  amounts.  How  much 
had  each  at  first  ? 

II. 

10.  Two  couriers,  A  and  B,  set  out  from  the  same 
place  and  travel  along  the  same  road  in  the  same  direc- 


174         CLEARING  EQUATIONS  OF  FRACTIONS. 

tion,  A  starting  8  hours  before  B.  B  rides  at  the  rate  of 
8  miles  an  hour,  and  A  at  the  rate  of  6  miles.  How  far 
will  each  have  travelled  when  B  has  overtaken  A  ? 

11.  A  and  B  find  a  sum  of  money.  A  takes  $2.40  and 
1/6  of  what  is  left;  then  B  takes  $3.52  and  1/7  of  what  is 
left;  and  they  find  they  have  taken  equal  amounts.  What 
was  the  sum  found  and  what  did  each  take  ? 

12.  A  fox  is  pursued  by  a  greyhound,  and  has  60  of 
her  own  leaps  the  start.  The  fox  leaps  three  times  while  the 
greyhound  leaps  twice,  but  the  hound  goes  as  far  in  3  leaps 
as  the  fox  does  in  7.  How  many  leaps  does  each  make 
before  the  hound  catches  the  fox  ? 

13.  A  hare  takes  4  leaps  to  a  greyhound's  3,  but  two 
of  the  hound's  leaps  are  equivalent  to  three  of  the  hare's. 
The  hare  has  a  start  of  50  of  her  leaps.  How  many  leaps 
must  the  hound  make  to  catch  the  hare  ? 

14.  A  man  and  a  boy  agreed  to  do  a  piece  of  work  for 
$5.25,  the  boy  to  receive  1/2  as  much  per  day  as  the  man. 
When  2/5  of  the  work  was  done  the  boy  left,  and,  in  con- 
sequence, it  took  the  man  1^  days  longer  to  complete  the 
work  than  it  would  otherwise  have  done.  How  much  did 
each  receive  per  day  ? 

15.  In  a  mixture  of  spirits  and  water,  half  of  the 
whole  plus  25  gallons  is  spirits,  and  a  third  of  the  whole 
minus  5  gallons  is  water.  How  many  gallons  are  there  of 
each  ? 

16.  A  garrison  of  1000  men  was  provisioned  for  60 
days.  After  10  days  it  was  reinforced,  and  from  that  time 
the  provisions  lasted  only  20  days.  What  was  the  number 
of  the  reinforcement  ? 

17.  A  laborer  was  engaged  for  36  days  on  condition 
that  he  should  receive  2s.  6d.  for  every  day  he  worked  and 
should  forfeit  Is.  6d.  for  every  day  he  was  idle.     At  the 


CLEARING  EQUATIONS  OF  FRACTIONS.        175 

end  of  the  time  he  received  58  shillings.      How  many  days 
did  he  work  ? 

18.  At  a  cricket  match  the  contractor  provided  dinner 
for  24  persons,  and  fixed  the  price  per  plate  so  as  to  gain 
12i  per  cent  upon  his  outlay.  Three  of  the  cricketers  were 
absent.  The  remaining  21  paid  the  fixed  price  for  their 
dinner,  and  the  contractor  lost  1  shilling.  What  was  the 
price  per  plate  ? 


CHAPTER  XYI. 
BADICALS  AND  SURDS. 

125.  Rational  and  Irrational  Numbers. — A  numerical 
quantity  which  can  be  exactly  expressed  as  an  integer  or  a 
fraction  whose  numerator  and  denominator  are  integers  is 
called  a  commensurable  or  a  rational  number,  and  one 
which  cannot  be  so  expressed,  an  incommeyisurdble  or  an 
irrational  number. 

126.  Radicals. — Any  algebraic  expression  which  con- 
tains a  factor  under  a  radical  or  other  root  sign  is  called  a 
radical  expression,  or  simply  a  radical^  and  the  factor 
under  the  root  sign  is  called  the  radical  factor. 

Any  algebraic  expression  which  contains  no  radical 
factor  is  called  a  rational  quantity. 

To  rationalize  an  expression  is  to  free  it  of  radical  or 
other  root  symbols. 

127.  Surds. — A  surd  is  an  incommensuratle  root  of  a 
commensurable  number.  In  other  words,  it  is  the  root  of 
an  arithmetical  number  which  can  be  found  only  approx- 
imately. 

While  every  surd  is  an  incommensurable  number,  there 
are  many  incommensurable  numbers  which  are  not  surds, 
or  due  to  any  finite  combinations  of  surds.  As  examples 
of  these  we  have  3.1415926  .  .  . ,  the  ratio  of  the  circum- 
ference to  the  diameter  of  a  circle,  and  2.7182818  .  .  . ,  the 
base  of  the  natural  or  Napierean  system  of  logarithms. 

176 


RADICALS  AND  SURDS.  177 

A  radical  expression  which  cannot  be  freed  from  root 
symbols  is  called  an  irrational  or  surd  expression,  or 
simply  a  surd.  The  symbol  of  a  surd  is  \^n,  in  which  n 
denotes  any  positive  integer,  and  a  any  integral  algebraic 
expression. 

A  surd  may  be  expressed  as  a  radical  quantity,  but 
every  radical  quantity  is  not  a  surd.  Thus,  V'6,  r5  are 
surds,  but  VT,  Vs  are  not  surds.  The  expression 
V  2  +  V2  is  not  a  surd  according  to  definition. 

128.  Imaginary  Quantities. — Since  no  even  combina- 
tion of  negative  factors  can  produce  a  negative  product,  an 
even  root  of  a  negative  quantity  is  called  an  imagiiiary 
quantity.  Thus,  V—  2,  V-^,  ^/-a  are  imaginary  quan- 
tities. 

The  value  of  the  expression  \^a  will  be  real  or  imag- 
inary  according  to  the  values  assigned  to  n  and  a.  It  will 
be  imaginary  when  n  is  even  and  a  is  negative.  In  all 
other  cases  the  value  will  be  real. 

When  ^  is  a  perfect  wth  power,  \^a  is  rational  and  in 
all  other  cases  irrational  or  surd. 

129.  To  Express  a  Rational  Quantity  as  a  Radical. — 

Any  rational  quantity  may  be  expressed  as  a  radical  by 
first  raising  it  to  the  power  indicated  by  the  index  of  the 
radical  and  then  placing  it  under  the  radical  sign. 

e.g.  4  =  Vie",     3  =  f27^ 

130.  Orders  of  Radicals. — A  radical  is  said  to  be  of  the 
first,  second,  or  nth.  orders  according  as  its  index  is  1,  2, 
or  n. 


178  RADICALS  AND  SURDS. 


EXERCISE  LXXIX. 

Express   the  following  quantities   as    radicals  of  the 
second  order: 


I. 

1. 

m. 

2.     n. 

3.     3a. 

4. 

hob. 

5.      7^^ 

6.     6a:^y^ 

7. 

1/4A. 

II. 

8. 

i/3ay. 

5aV      . 

10.     a-\-b. 

11. 

x-y. 

12.     3«2  +  7. 

Write  the 

following  as  radicals  of  the  third  order: 

13. 

X. 

I. 
14.     Sa^x. 
II. 

16.     1/3«V. 

16. 

x^h. 

17.     a- 3. 

18-     -rr- 

131.  Arithmetical  Boots. — We  have  already  seen  that 

Va/^  has  two  values,  +  a  and  —  a;  also,  that  Va  has  two 

values  which  differ  only  in  sign,  one  being  positive  and 

the  other  negative.     In  higher  algebra  it  is  shown  that 

Va  has  three  values,  one  of  which  is  real  and  the  other 

two  imaginary;  also,  that  Va  has  9i  values,  one,  or  at 
most  two,  of  which  may  be  real,  and  the  others  imaginary, 
and  that  when  there  are  two  real  roots  they  will  differ  only 
in  sign. 

When  a  root  symbol  is  placed  before  a  number  it  de- 
notes the  arithmetical  root  only,  but  when  placed  before 


RADICALS  AND  SURDS.  179 

an  algebraic  expression  it  denotes  one  of  the  roots.     Thus 
Va  has  two  values  either  of  which  is  denoted  by  the  sym- 
bol, but  V^  is  supposed  to  denote  only  the  arithmetical 
root,  unless  it  is  written  ±  V^. 

In  the  demonstrations  in  the  present  chapter  the  sym- 
bol Va  in  all  cases  must  be  taken  in  a  restricted  sense, — to 
mean  the  real  root  of  a  whose  sign  is  the  same  as  the  essen- 
tial sign  of  a.     Thus  Va"  must  be  taken  to  mean  a,  and 

Va  to  mean  the  one  real  root  of  a  which  has  the  same  sign 
as  a.  The  theorems  established  in  this  chapter  do  not 
necessarily  apply  to  other  real  roots  than  the  one  specified 
above,  or  to  imaginary  roots. 

In  this  chapter  it  is  assumed  that  the  associative,  dis- 
tributive, commutative,  and  index  laws,  which  have  been 
established  for  integers,  and  applied  to  rational  algebraic 
expressions,  also  apply  to  surds. 

132.  Theorem  I.  The  product  of  the  same  roots  of  ttuo 
factors  is  equal  to  that  root  of  the  product  of  the  factors. 

By  definition  \a  used  n  times  as  a  factor  will  give  a  as 
a  product. 

.-.    C^ay  =  a. 
Similarly,      (  VhY  =  h,     and     (  "x/abY  =  ah. 
But      (  y~a  X  "i^lY  =  (  VaT  X  (  ^Y  =  ah, 
and  (  VahY  =  ah. 

/.     iVaxny=(VabT.     (Why?) 
/.     -\^axVI=  'i^^.     (Why?) 

Cor.  The  product  of  the  same  roots  of  any  number  of 
factors  is  equal  to  that  root  of  the  product  of  those  factors. 


180  RADICALS  AND  SURDS. 

Note. — It  should  be  borne  in  mind  that  \^~a,  taken 
arbitrarily,  x  V~b,  taken  arbitrarily,  does  not  =  Vab,  taken 
arbitrarily.  Thus  the  negative  root  of  2  multiplied  by  the 
positive  root  of  3  does  not  equal  the  positive  root  of  6. 

The  equation  Va  X  yb=^  Vah  is  true  when  the  mean- 
ing of  the  symbols  is  restricted  as  in  131.  It  is  also  true 
that  any  one  of  the  n  roots  of  a  multiplied  by  any  one  of  the 
n  roots  of  i  will  be  equal  to  some  one  of  the  n  roots  of  ab. 

133.  It  follows  from  Theorem  I  that,  when  the  quantity 
under  the  radical  sign  can  be  separated  into  factors  one  or 
more  of  which  is  an  exact  power  of  the  order  of  the  root 
indicated,  the  product  of  the  indicated  roots  of  these  factors 
may  be  placed  as  a  factor  outside  the  radical. 

e.g.    l/l92  =  1^16  X  4  X  3  =  VU  X  Vlx  V^=  8  V^. 

f  864  =  \/Wx  8X4  =  f  2f  X  ^^8  X  ^4  =  6  1^4: 

134.  Pure  and  Mixed  Surds. — The  factor  without  the 
radical  sign  may  be  regarded  as  the  coefficient  of  the  radi- 
cal. 

A  ptire  surd  is  one  that  has  no  rational  coefficient 
except  unity. 

A  mixed  surd  is  one  that  has  a  rational  factor  other 
than  unity. 

A  surd  is  said  to  be  in  its  simplest  form  when  it  has  no 
rational  factor  under  the  radical  sign. 

EXERCISE  LXXX. 

Write  the  following  as  mixed  surds  in  their  simplest 
forms : 

I. 

4.      ^7357  5.      VWi^.  6.      ^5677 

7.      fl35. 


MABICALS  AND  SURDS.  181 


II. 


8.     V448.  9.      i/5632.  lo.      4/48a2J. 


11.      4/l25«V.  12.    VUlaH\ 


13.      l/4«&  +  8«^^  +  4a3^l   14.    1/I22;y  -  24a;y  +  12a^y\ 

A  mixed  surd  may  be  reduced  to  the  form  of  a  pure 
surd  by  raising  its  coefficient  to  the  power  indicated  by  the 
order  of  the  surd  and  placing  it  as  a  factor  under  the  radi- 
cal sign. 

e.g.  7  1/5  =  1/72^5  =  '^245. 

EXERCISE    LXXXI. 

Express  the  following  as  pure  surds: 
I. 

1.   '^Vn.         2.   41/13;  3.   6 1/7. 

4.     2  V^.  5.     4  \^6.  6.     6  t^4. 


7.     3^  Va  —  b.     8.     {x  +  ij)  V'Sx.      9.     'Sa(a  —  b)  Vbab. 

136.  Theorem  II.  The  quotient  of  the  same  roots  of 
two  qtiantities  is  equal  to  that  root  of  the  quotie?it  of  the 
two  quantities. 

Expressed  algebraically,  Va  -^  Vb  =  \^a  -^  b, 

(  V^^  ny  =  (  ;/ay  -  (  f^)"  =  a~.b. 
But  (  \^a  --  by  =  a^  b, 

•/.      (  4^«  --  \^by  =  (  f ^TT^)'.     (AVhy  ?) 
/.     i^a  -4-  yT  =  1^«T~^»     (Why  ?) 

Cor.     a  /  ~  =    ,,  _=     That  is,  any  root  of  a  fraction 


«  /a        Va      ,p 


182  RADICALS  AND  SURDS. 

may  be  indicated  by  placing  the  corresponding  radical  over 
the  fraction  as  a  whole,  or  over  its  numerator  and  denomi- 
nator separately. 

136.  Similar  and  Quadratic  Surds. — Similar  surds  are 
those  whose  radical  factors  are  identical,  e.g.  Vb,  3  Vb, 
are  similar  surds.     So  also  are  a  ^x  and  c  ^x. 

Surds  of  the  second  order  are  called  quadratic  surds. 

137.  Theorem  III.  The  product  of  two  similar  quad- 
ratic surds  is  a  rational  quantity. 

mVaXnVa  —  7mi  Va^  =  mna. 

The  product  of  the  coefficients  is  necessarily  a  rational 
quantity,  and  the  product  of  the  similar  radical  factors  is 
necessarily  the  square  root  of  a  perfect  square,  and,  there- 
fore, rational. 

138.  Theorem  IV.  The  product  of  two  dissimilar 
quadratic  surds  canyiot  le- rational. 

Let  ^a  and  ^h  be  the  surd  factors.  Since  the  surds 
are  dissimilar,  a  and  h  cannot  be  composed  of  the  same 
prime  factors,  and  hence  their  product  ah  cannot  be  com- 
posed of  square  factors  only.  Therefore  'fab  cannot  be 
rational. 

139.  Rationalizing  Factor.  —  Any  factor  which  will 
convert  a  radical  expression  into  a  rational  one  is  called  a 
rationalizing  factor. 

It  follows  from  Theorem  II  that  the  surd  factor  of  a 
pure  or  mixed  surd  is  a  rationalizing  factor. 

e.g.      l/5'xl^==5.         3  V3"x  |/3  =  3  X  3  =  9. 


h^a-hy.^a-h  =  h{a-V). 


RADICALS  AND  SURDS.  183 

140.  To  Reduce  a  Fractional  Radical  to  an  Integral 
Radical. — A  fractional  radical  may  be  reduced  to  an  inte- 
gral radical  with  a  fractional  coefficient,  by  writing  its  nu- 
merator and  denominator  each  as  a  separate  radical,  and 
then  multiplying  each  by  the  rationalizing  factor  of  the 
denominator. 

e.g.  ^5-= -^  =  -^£^=1/5^. 

/^  _  _V«  _   V^x  Vb  _   Vah  _-\^     

y  h~  Vb~  VbxVb~     b    -i^^^' 

EXERCISE  LXXXII. 

Reduce  the  following  to  integral  radicals : 
I. 
1.     VT/2.  2.     VT/b.  3.     Vy^. 

.  vm,       .  /|-        a.  /f±|. 

II. 


7. 


./x  +  4  ^x  —  5  a/6x  —  2 


/4:X  -  6  Jb  -  2x 


10.      f .  11.      V  .  12. 

—Zx-^1 L-^-4^  4 

141.  Addition  and  Subtraction  of  Radicals. —  Similar 
radicals  may  be  added  and  subtracted  by  combining  their 
coefficients  in  the  same  way  as  similar  rational  terms.  The 
common  surd  factor  must  be  written  after  the  coefficient 
resulting  from  the  combination. 

e.g.  The  sum  of  3  V5,  0  V5,  and  -  7  VE  is  6  VE. 

The  difference  of  3  V2  and  9  1^  is  -  6  1^2. 


184  RADICALS  AND  SURDS. 

Dissimilar  radicals  can  be  added  and  subtracted  only  by 
writing  them  one  after  another,  each  with  its  proper  sign, 
as  in  the  case  of  dissimilar  rational  terms. 

Thus,  Vy  added  to  Vx  =  Vx  +  V^,  and  never 
Vx  -{-  y,  unless  either  x  or  y  i^  zero. 

143.  Rule  for  Addition  of  Radicals. — To  add  surds  of 
the  same  order,  reduce  them  to  their  simplest  forms  and 
add  the  coefficients  of  the  resulting  surds  which  are  similar, 
and  write  those  which  are  dissimilar  after  one  another. 

e.g.       ^V^ -\-VT^-{-%^/n  -  V^-\-^Vb 

143.  Rule  for  Subtraction  of  Radicals. — To  subtract 
two  radicals  of  the  same  order,  reduce  them  to  their 
simplest  form,  and  then,  if  they  are  similar,  subtract  their 
coefficients,  and  if  they  are  dissimilar,  write  them  one 
after  the  other  with  the  proper  sign  between. 

e.g.    From  3  Vb  take  2  Vv^. 

=  3  V5  -  10  1/5"=  -  7  y^. 
From  3  \^  take  2  VSO. 

144.  Addition  and  Subtraction  of  Radicals  of  Differ- 
ent Orders. — Radicals  of  different  orders  can  be  added 
and  subtracted  only  by  writing  them  one  after  another 
with  the  proper  signs  between. 

EXERCISE   LXXXIII. 

Find  the  sum  of  the  following  sets  of  radicals  : 
I. 

1.  4/18',    -^32,    1^50,    and  Vn. 

2.  2  VS,  3  /50,  and  6  4^18. 


RADICALS}  AND  8UBD8.  185 


3.  i/3/5,    ^1/15,    and   ^15/49. 

4.  2/3^^279;   l/ev'ITse,   and   d/b\/yZ2. 


5.  xVVZa%    2a^V27x\    3«  V48«V,    and  |/75aV. 

II. 

6.  2V3,    1/2  Vl2,    4^27,   and    4/T27I6. 

7.  I'54^^    7«y'2^^    and   8b  \^2a^^ 
7nn 


and 


y   (n  —  s 


71  —  S  \     {^  ~  ^)  ^^  ~ 

EXERCISE  LXXXIV. 
I. 

1.     From  2  1/320"  subtract  3  I^SO. 


2.     From  «  |/646f^Z>4  subtract  i  VWda%. 


3.     From  Va%  +  2a52  _^  J3  subtract  Va%  -  2aW  +  ¥. 


4.    From  |/2a34-46j2^+2«J2  subtract  ^Iw^-^d'l^laV^. 
II. 


5.    From  2/3  1^2/9  +  3/5  1^3/32  subtract  1/6  Vl/36. 


6.     From  l/289a3j  subtract  3  VlUa^, 


7.  From  2  l/Sc^  +  5  4/72c3  subtract  7c  VlSc  +  I^SOc^^, 

8.  From  (c  —  :c)  Vc'^  —  x^  subtract  a  /  -^ — . 

\J    c  —  X 

145.  Multiplication  of  Radicals  of  the  Same  Order. — 

To  multiply  together  two  radicals  of  the  same  order, 
multiply  together  their  coefficients  for  the  new  coefficient, 
and  the  quantities  under  the  radical  sign  for  the  new 
radical. 


186  RADICALS  AND  8UBD8. 


EXERCISE  LXXXV. 

Perforin  the  following  multiplications  and  reduce  the 
results  to  the  simplest  form : 


1.  3  f^  X  2  Vm,  2.     7^2/81  X  3/2  \/y^^. 

3.  4  i/l2  X  3  V%  4.     ^^1727  X  3/4  ^12. 

5.  5  Vc^  X  1/2  V^bix.     6.     ^  t^2a^  X  a  ^Sab. 

7.  (2  V2  -  3  V3  +  4  4/5)  X  (3  4/5  +  4  1^3). 

8.  (3V5-4:  V2)  (24/5  +  3  4/2). 

9.  (  4/7"+  5  4^)  (2  1^  -  4  VS). 

10.  (  4/2  +  4/3  -   4^)  (  4^+  4/3  +  4^). 

11.  (3  4/^ -  2  ^)  (2  4^+  3  4/^). 

12.  Multiply  4/f  +  9  by4^  -  6. 

13.  Square  4^5  +  3. 

14.  Multiply  4^  -  6  by  4^  ~  8. 
16.  Square  VY  —  5. 

16.  Multiply  t^  +  4  by  V^  +  3. 

17.  Square  Vx  -{-  9. 

18.  Multiply  Vx  -{-  Q  hj  Vx  —  5, 

19.  Square  Vx  —  Vs. 

20.  Multiply  4^  +  4/7  by   VE  -  VY. 

21.  Square  tY  +  4^8. 

22.  Multiply  4^2;  +  5  by  4/:^  —  8. 


RADICALS  AND  SURD8.  187 


23.     Square  Vx  —  4=  -\-  Vx  -\-  6. 


24.     Multiply  4/a;  +  7  by  Vx  -  7. 


25.     Square  l^a;  —  3  +  Vo;  -|-  3. 


II. 


26.  Multiply  3xVa  —  ij  by  5x  Va  —  7. 

27.  Square  2a  VQx-\-2  Vb. 


28.  Multiply  5aVx-{-7  by  7Z>  4/3;  +  7. 

29.  Square  d  Vx  -{-  6  —  4:Vx  —  7. 


30.  Multiply  7^/^  Va;  —  4  by  9^  Vx  -f  4. 

31.  Square  3a  Va  -\-  3  -\-  5a  Va  —  5. 


32.  Multiply  Vx  —  4:  —  5  by  4/a;  —  4  +  5. 

33.  Multiply  1^2'  +  8  +  VQ  by  i^o;  +  8  -  Vq. 


34.  Multiply  Va;  -  5  +  i^a;  4-  8  by  4/3;  -  5  -  1^2;  +  8. 

35.  Multiply 


3  Vx  -\-  6  -^  4:  Vx  -{-  5  hy  3  Vx  -\-  6  -  4:  Vx  -i-  6. 
36.     Multiply 


3a^x  Vx-S  ~  5xWx-\-7  by  3  A  Vx  -  S  +  6x^  Vx  +  7. 

146.  Simple,  Compound,  and  Conjugate  Radicals.— A 

smjy/e  radical  expression  is  one  which  contains  only  one 
term,  and  a  compou?id  radical  expression  is  one  which  con- 
tains more  than  one  term. 


Thus,  Vx,  V  a-\-  X,  a  Vab,  (a  -\-  b)  Vx  -\-  4,  are  simple 
radicals,    a  -\-  Vx,  Va  +  Vx  -\-  b,  are  compound  radicals. 
Two  binomial  quadratic  radicals  which  have  the  same 


190  RADICALS  AND  SURDS. 

^  4.      ^    ^    . 

5  +  2^ 

^2 


9  +  2V1T 

.. 

a^Va'-  b^ 

II. 

V3-\-a^-  V3 

-«2 

0 

Vd-i-a^-\-  V'd 

-a^-         '• 

V5  +  a;2  4-2 
3  + 

4/6 

6. 


^a^  ^y^-y 


2Vx-i-3-^SVx-3 
2  V^"+3  -  3  Vx'^^' 


6V3  -2  Vi2  -  VS2  +  VW 
Divide  the  following  radicals  at  sight : 
I. 
11.      VT8"by  Vq.  12.      I^2rby  \^. 

13.     12V35by3i^.  14.     a^  Vb^hj  a"^  Vb. 


15.      Vx^  -  49  by  Vx  +  7.  le.     Ya^^  -  8  by  i^a;  -  2. 


3/ 


17.      i^x^  4-  27  by  ya;2  _  3a;  +  9. 

II. 


18. 

Vx^-\-2x-  15  by  Vx-}-6: 

19. 

Vx"-  13a; +  42  by  Vx  -  Q. 

20. 

Vx^  -x-72  by  Vx  +  8. 

21. 

V6x^  +  17a;  -  14  by  V2x  +  7. 

22, 

V6x  -  2a;  -  7  by  fa;  +  1. 

RADICALS  AND  SURDS.  191 

Divide  the  following  radicals  by  first  expressing  the 
division  in  the  form  of  a  fraction  and  then  rationalizing 
the  denominator. 

I. 

23.  29  by  11  +  3  Vl. 

24.  17  by  3  i^  +  2  Vd. 

25.  3|/2  -  1  by  3|/2  + 1. 

26.  2  i^+  7  1/2  by  5V3-4.V2. 

II. 

27.  ^x  —  Vxy  by  2  Vxy  —  y. 

28.  (3  +  Vl)(  1/5  -  2)  by  5  -  Vb, 

Va         ,       Va-\-  Vx 
by 


2  1^15"+ 8        8  1/3  -  6  V5 
5  H-  VTS"  ^^  5  V3  -  3  VS  * 

150.  Theorem  IV.  The  71th  power  of  the  root  of  any 
quantity  is  the  same  root  of  the  nth  power  of  the  quantity, 
71  and  the  index  of  the  root  both  being  positive  integers. 

1°.  When  the  index  of  the  root  is  the  same  as  the 
exponent  of  the  power. 

By  definition,  (^ya)"  =  af 

and  r  «"  =  a. 

,'.      (i/aY  =   te 

2°.  When  the  index  of  the  root  is  not  the  same  as  the 
exponent  of  the  power. 

(ni    —  v  )n  n 

Va")    =  a  , 


19S  RADICALS  AND  SURDS. 

■^^^^  yi^Vaj)    means  that  Vfl^  is  to  be  used 

mn  times  as  a  factor,  and 

(( "Va  )")    means  that  "Va  is  to  be  used 
mn  times  as  a  factor. 

...  ((?«)•")" =((i^)"r. 

But  ((?«)")"  =  «». 

.-.  ((r«)r=«"- 
...  ((v«)T=("v;?r. 

151.  Theorem  V.  The  mth  root  of  the  nth  root  of  a 
quantity  is  equal  to  the  mnth  root  of  the  quantity. 

Imf  n  _\m 

By  definition,    \V  Va)    =  V^. 

...     {(VWrY=(^'a)''=a. 
Also,  ("'f^)»»=«, 

and  {(Vl^TY=(Vl^r. 

152.  To  Change  Radicals  from  One  Index  to  Another. 
— It  follows  from  Theorems  IV  and  V  that  a  radical  may 
be  changed  from  one  index  to  another  by  multiplying  both 
the  index  of  the  radical  and  the  exponent  of  the  quantity 
under  the  radical  by  the  number  which  will  produce  the  re- 


RADICALS  AND  SURDS.  193 

quired  index.  For  the  former  of  these  operations  would 
extract  a  root  of  the  radical  quantity,  and  the  latter  would 
raise  it  to  the  corresponding  power,  and  these  two  opera- 
tions would  neutralize  each  other. 

e.g.  \/a^^"''\/^^='\/^. 

To  change  radicals  of  different  orders  to  those  of  the 
same  order  with  the  smallest  possible  indices,  multiply  each 
index  by  the  quotient  obtained  by  dividing  the  least  com- 
mon multiple  of  all  the  indices  by  that  index  and  raise  the. 
quantity  under  the  radical  sign  to  the  corresponding  power. 
This  will,  of  course,  make  the  index  of  each  radical  the 
least  common  multiple  of  all  the  indices. 

e.g.  Reduce  4^5,   Vd,  and  r  2  to  radicals  of  the  same 
order  with  the  smallest  possible  index. 
The  L.  0.  M.  of  2,  3,  and  4  is  12. 


2X6 


1/5=  T5«=  ?15635. 

163.  Multiplication  and  Division  of  Radicals  of  Different 
Orders. — Radicals  of  different  orders  may  be  multiplied  to- 
gether by  first  reducing  them  to  the  same  order  and  then 
multiplying  together  their  rational  and  their  irrational 
factors. 

Similarly,  radicals  of  different  orders  may  be  divided  by 
each  other,  by  first  reducing  them  to  radicals  of  the  same 
order  and  then  dividing  their  integral  and  radical  factors. 

EXERCISE  LXXXIX. 

1.     Reduce  r  10,   1^5,  and  r  11/12  to  a  common  index. 


2.     Reduce   Va  +  h,    Va  —  b,  and   Vd^  +  a;^  to  a  com- 
mon index. 


194  RADICALS  AND  SURDS. 

3.  Multiply   V^  by   Vb. 

4.  Multiply  \/yYhy  VyI. 
6.    Divide  Va^  by  \d^. 

6.  Divide  2  V'Zac  by  t^4^c^. 

7.  Divide  1/2  ^273  by  1/3  I'lTsT 

164.  Radical  Equations. — An  equation  which  contains 
radicals  is  called  a  radical  equation.  Such  equations  are 
solved  by  first  clearing  them  of  radicals,  or  rationalizing 
them.  If  the  equation  contains  fractions  it  should  be 
cleared  of  them  first  of  all. 

In  the  case  of  a  quadratic  radical  equation,  after  it  has 
been  cleared  of  fractions,  it  is  best  to  transpose  all  the 
terms  into  the  left-liand  member  and  place  this  equal  to 
zero.  Each  member  should  then  be  multiplied  by  the  con- 
jugate of  the  first. 

If  the  first  member  contains  more  than  two  terms,  they 
should  first  be  collected  into  a  term  and  an  aggregate,  or 
into  two  aggregates,  and  the  terms  arranged,  if  possible,  so 
that  the  aggregate  shall  contain  no  radical.  Multiplying 
then  by  the  conjugate  expression  will  square  each  of  the 
terms  or  aggregates,  and  place  the  minus  sign  between  the 
squares  obtained,  and  the  result  will  be  rational.  If  either 
aggregate  contains  a  radical,  the  result  of  the  first  squaring 
will  be  irrational.  In  this  case  a  new  pair  of  aggregates 
must  be  formed  and  the  operation  must  be  repeated. 

e.g.  1.  Vx  —  Q  -4=9. 

Transposing,  we  get 

V^^^Q  -  13  =  0. 

Multiplying  by  the  conjugate  expression  Vx  —  ^  -j-  13, 
we  get 


RADICALS  AND  SURDS.  195 

X-  6  -  169  =  0, 

2.  V^^^  +  2  1^-5  =  3. 

Transposing,  we  get 

V^^^d  +  2  1^  -  8  =  0. 
Writing  this  as  the  sum  of  two  aggregates,  thus, 

Vl^^d  +  (2  ^^  -  8)  =  0, 
and    multiplying     this     by     the     conjugate    expression 
V¥x'^3  -  {2Vx  —  8),  we  get 

4:X-S  -4:X-{-32Vx-64:  =  0. 
Collecting,  we  get 

32  1^-67  =  0. 

Multiplying   again  by  the  conjugate  32  V^  +  67,  we 
get 

1024a;  -  4489  =  0. 

EXERCISE  XC. 

Solve  the  following  radical  equations : 

I. 


l^Vx-  5  =  3.  2.    V4:X-7  =  5. 


3.     7  -  Vx-4:  =  3.  4.    2  |/5x  +  4  =  8. 


6.     VSa;  -  1  =  2  Va;  +  3. 


6.     2  |/3  -  7a;  -  3  VSx  -  12  =  0. 


7.  Vx-\-25  =  l  -\-  Vx. 

8.  V8a;  +  33  -  3  =  2  V2x. 


196  RADICALS  AND  SURDS, 


9. 

Vx- 
10  - 

Vx- 

V^-^Vx 

=  5. 

10. 

-  V25  4-  9 
-4  +  3  = 

x^'dVx. 

11. 

Vx  +  11, 

12.     V9a;  -  8  =  3  |/a;  +  4  -  2. 


II. 


13.     Vx  -\-  Aab  =  2«  +  Vx, 


14.  Vx  +  V4:a  -\-x  =  2Vb-\-x. 

15.  "^x^  +   ^4:r2  +  x  +  Vfx^'-\-'V2x  =  1  ^  x. 


16.  '^«  +  Vax  —  Va  —   Va  —  Vax. 

17.  Vx  -\-  Vax  —  a  —  1. 


18. 


19. 


21. 


Vs-c 

+     '' 

.=  Vbx^  6. 

'    Vsx  +  (j 

4^- 

_  237  - 

4  + 

10:?; 

4/^   • 

Va- 

0^ 

—   T-   — 

^ 

Va- 

-  a; 

Vx- 

a 

7^+3  = 

:c-4 

i^^+2  * 

Vx  -\-Va-x. 

V,T- 

-!/«-:?;  = 

155.  Reduction  of  Radical  Equations  by  Rationaliza- 
tion.— When  a  radical  equation  contains  but  one  radical 
fraction,  it  is  often  best  to  rationalize  the  denominator  of 
that  fraction  before  clearing  of  fractions. 


RADICALS  AND  SURDS.  197 


Va-\-x-\-Va  —  X 

e.g.  '_    — , =  h. 

ra-\-x  —  Va  —  x 

Rationalizing  the  fraction,  we  get 


2a  4-  2  Va^  -  x^        ,                a  4-  Vif  -  x^ 
=  b,     or     =  h. 

2x  X 

Clearing  of  fractions  and  transposing,  we  get 


a  —  bx  -{-  Va^  —  x^  —  0. 

Multiplying  by  the  conjugate,  we  have 

«2  _  2abx  +  b^x^  -  «2  -\.x^  =  0, 

or  (Z>2  +  l)a;2  =  2abx. 

.-.     {b^^l)x  =z2ab. 
_     2ab 
'''     "^  ""F+~l* 

EXERCISE  XCI. 

Solve  the  first  four  of  the  following  equations  by  ration- 
alizing the  denominator: 

I. 


V3  +  X  A-  V'S  -  X       ,  V6  +  ^  +  |/6 

—  =  4-.    2  ^ — _= 

\/S  -^  X  —  V3  —  X  V6  i-  ^-  —  V6 


Vx-^^-{-Vx        ^  Vx^^-\-Vx       _ 

•  ;=  =  5.  4.  ,  -j^—  lU. 

|/a;  _^  4  _  ^x  Vx-\-Q  —  Vx 

II. 


l^a;  +  « 

+  i^ 

Vx  +  fl'- 

-|/^ 
-|/^ 

1^2  +  2: 

7 

V2-\-x 

+  Vx 

12' 

6. 


1^2  +  a;  -  l^a;   _  5_ 
V2~-^  -i-  Vx        9  ' 

V2-\-x  —  Vx  _b_ 
V2-\-x-]-  Vx  ~  ^ 


CHAPTER  XVII. 

THE  INDEX  LAW. 

156.  Meaning  of  Fractional  Exponents. —  It  has  been 
shown  that,  when  7n  and  n  are  positive  integers, 

or  xa''  =  or  +  ''.  (1) 

Also  as  a  corollary  to  this,  when  m>  n, 

or  ^  a""  =  ar-"". 
And  as  a  consequence  of  (1)  it  has  been  shown  that 

{ary  =  «»»«  =  (a^^)^,  (2) 

and  («^)"  =  a'^b^  (3) 

These  three  laws  follow  from  the  definition  that  an  ex= 
ponent  denotes  the  number  of  times  a  quantity  is  employed 
as  a  factor. 

The  law  expressed  by  equation  (1)  is  known  as  the  In- 
dex Law. 

The  definition  of  an  exponent  becomes  meaningless  if 
the  exponent,  or  index,  be  other  than  a  positive  integer. 

The  spirit  of  algebra  is  to  generalize,  and  the  use  of 
indices  cannot  be  restricted  to  the  particular  case  of  inte- 
gers, but  it  must  be  extended  to  the  case  of  fractional, 
zero,  and  negative  indices.  All  of  these  indices  must  be 
governed  by  the  index  law,  and  they  must  be  interpreted 
in  accordance  with  this  law. 

We  will  proceed  first  to  find  the  meaning  of  a  fractional 
index  in  which  both  numerator  and  denominator  are  positive 
integers. 

198 


THE  INDEX  LAW.  l90 

Let  this  index  be  denoted  by  - . 

q 

Since  the  equation  a"^  .  a"  =  (fi  +  n  jg  ^^  -^^  ^^^^  ^^^  ^y[ 
values  of  m  and  n,  we  may  replace  each  by  — .  We  then 
have 

P  P  2p 

and  multiplying  each  member  by  a**,  we  get 
and  so  on  up  to  q  factors,  when  we  should  have 

P_  P_  P_  OP 

a*^  ,  a*^  .  a*^ .  .  .  .  q factors  =  a'^  =  a^» 

.  •.     (aT)  =  a». 

Therefore,  by  taking  the  g'th  root  of  each  member,  we 

have 

p_         

p_ 
or,  in  words,  ft  **  is  equal  to  ^'the  5'th  root  of  a  to  the  ^th 

power." 

If  ^  =  1,  we  should  have 

1 
or  ft"  is  equal  to  the  nth  root  of  a. 

For  the  present  the  meaning  of  the  symbol  «»»  must  be 
restricted  to  the  real  nth.  root  of  a  whose  sign  is  the  same 
as  the  essential  sign  of  a,  or  to  what  may  be  called  the 
arithmetical  root  of  a.  If  this  strict  limitation  is  departed 
from,  we  are  led  to  various  paradoxes. 

e.g.  By  the  interpretation  of  fractional  indices 


200  THE  INDEX  LAW. 

But  icV2  z=  x^, 

which  is  right  if  we  take  x^f^  to  stand  for  the  positive  value 
of  V^',  but  leads  to  the  paradox  x^  =  —  x^  if  we  admit 
the  negative  value. 

Again,  according  to  the  index  law, 

and  (9V2)2  ^  (92)1/2^ 

or  (±3)2  =±9, 

or  9  =  ±  9, 

if  both  values  are  admitted. 

157.  Meaning  of  Zero  Exponent. — Since  al^ ,  a^  =  oT -^ '^ 
is  to  hold  for  all  values  of  7n  and  n,  we  may  replace  7n  by 
zero.     We  then  have 

aO.«"  =  ««  +  "  =  a". 

Therefore,  by  dividing  each  member  by  a",  we  get 

0  ^"  1 

a" 
Therefore  a  quantity  with  zero  index  is  equal  to  1. 

158.  Meaning  of  Negative  Exponents. — Since  a^ .  «"= 
^m  +  n  ^g  ^Q  ^^o\^  for  all  values  of  m  and  ?^,  we  may  replace 
m  by  —  n.     We  then  have 


Therefore  by  dividing  each  member  by  a"  we  get 

fi~^  — 

""      -a--  or' 
Also,  dividing  each  member  by  a~"  we  get 


a"  = 


a~"      a 


THE  INDEX  LAW.  201 

Hence  a  quantity  with  a  negative  exponent  is  equal  to 
the  reciprocal  of  the  same  quantity  with  the  corresponding 
positive  exponent. 

Cor.  Any  factor  may  he  transposed  from  the  denom- 
inator to  the  numerator  of  an  expression,  and  the  reverse, 
hy  simply  changing  the  sign  of  its  exponent. 

159.  The  Index  Law  holds  for  all  Rational  Values  of  w 
and  n. — Now  that  we  have  found  what,  in  accordance  with 
the  index  law,  indices  must  mean  for  all  rational  values  of 
m  and  n,  we  must  show  that,  with  these  meanings,  the 
three  laws 

ft'"  .  a«  =  «»"+«,  (1) 

{cd^Y  =  a^^^  (2) 

and  («^J)"  =  a"^>"  (3) 

must  hold  for  all  rational  values  of  m  and  n. 

I.  To  show  that  a"^  .  a""  =  «""+*  for  all  rational  values  of 

w  and  n. 

t)  r 

1°.  Let  m  and  n  be  any  fractions  —  and  -,  in  which j3, 

q,  r,  and  s  are  positive  integers. 

Then        a^'^ .  a''"  =  ^aP  •  f  a^,  by  definition; 

=  YaP'  .7a^,  by  152; 


y^ps+rq^  by  132; 


pa+rq 


=  a  ^^  ,  by  definition 

—  ^plq+rfs  __  ffti+n 

If  either  m  or  7i  is  a  positive  integer  while  the  other  is 
a  fraction  with  positive  integers  for  its  numerator  and  de- 
nominator, the  integer  may  be  expressed  in  a  fractional 
form,  and  the  demonstration  just  given  will  hold. 


202  THE  INDEX  LAW. 

We  know  already  that  the  law  holds  when  m  and  7i  are 
positive  integers.     Therefore 

for  'all  positive  rational  values  of  m  and  n, 

2°.  Let  m  and  n  be  essentially  positive,  either  fractions 
or  integers. 

11  1 


Then 


m      fi-n 


a  ".a 


or      or       a' 


by  definition. 

And  if  w  —  w  be  positive. 


and  or  .  a'""  .«"  =  «"*.  —  .  «"  =  a"». 

a" 

. *.     a    .  a-"",  a""  =  a"*""  .  a". 
Hence  if  m  —  w  be  negative,  that  is  ?^  —  m  be  positive. 

Therefore  for  all  rational  values  of  m  and  n 

«,«*.  «"■=  «*"  +  ". 

Cor.  Since  «*""''.  «"  =  r?""  for  all  rational  values  of 
m  and  n,  it  follows,  by  dividing  both  sides  by  a^,  that 

^m  _^  ^n  -.  f^m-n  f qj,  ^jj  ^atlonal  valucs  of  m  and  7i. 

II.  To  prove  that  («"*)"  =  a"*"  for  all  rational  values  of 
m  and  n. 

1°.  Let  m  have  any  value  whatever,  and  let  7i  be  a 
positive  integer. 


THE  INDEX  LAW.  203 

Then,  by  definition, 

{cry  =  or  .or  '  a"^  .  -  'to  n  factors 

=  oT''. 

2°.  Let  m  have  any  value  whatever,  and  let  ti  be  a 
fraction  -,  in  which  p  and  q  are  positive  integers. 

Then  {aTy^  =  ^{oTY,  by  definition; 
=  ^a-^byII,  1°; 

mp 

=  tt  3  ,  by  definition; 

=  a"*' 7  =  «"»". 
3°.  Let  n  be  any  rational  negative  quantity  and  =  —i?. 

ThenK)-^=  (^  =  J^p  =  «-""• 

We  know  already  that  the  law  holds  when  m  and  n  are 
positive  integers. 

Hence  for  all  rational  values  of  ?n  and  n 

III.  To  prove  (aby  =  «"Z>"  for  all  rational  values  of  7i. 
1°.  Let  n  be  any  positive  rational  quantity  which  may 

be  denoted  by  a  fraction -,  in  which  p  and  5'  are  positive 

integers. 

Then  (aby  =  (ab)^  =  ;{/{abY,  by  definition; 
=  ^a^b^,  by  (2). 


202  THE  INDEX  LAW. 

We  know  already  that  the  law  holds  when  m  and  n  are 
positive  integers.     Therefore 


for  "all  positive  rational  values  of  m  and  7i. 

2°.  Let  m  and  n  be  essentially  positive,  either  fractions 
or  integers. 

Then    a""* .  a""  =  J-  .  1-  =  -^^  =  a-"*-*^. 


by  definition. 

And  if  m  —  ^  be  positive. 


and  «*".«-".«"  =  «"*.  —  .  «"  =  a™. 

a"* 

.-.     a    .  a-"",  a""  =  a"*-",  a". 

.  •.     a"^  .  <?-"  =  «"*-'*. 

Hence  if  m  —  '^  be  negative,  that  \^  n  —  m  be  positive. 

Therefore  for  all  rational  values  of  m  and  ?^ 

aJ^  .  «"  ■=  «»"  +  ". 

Cor.  Since  «"'"''.  «"  =  r?""  for  all  rational  values  of 
m  and  n,  it  follows,  by  dividing  both  sides  by  0",  that 

^m  _^  ^n  -.  ^m-n  f ^j.  ^|j  national  values  of  m  and  /i. 

II.  To  prove  that  («"*)"  =  «"*"  for  all  rational  values  of 
m  and  ^z. 

1°.  Let  m  have  any  value  whatever,  and  let  71  be  a 
positive  integer. 


THE  INDEX  LAW.  203 

Then,  by  definition, 

(ary  =  or  ,ar  ,  or  .  ,  .to  n  factors 
=  a 
=  oT'', 

2°.  Let  m  have  any  value  whatever,  and  let  w  be  a 
fraction  -,  in  which  p  and  q  are  positive  integers. 

Then  {aTy^  =  ^(oTY,  by  definition; 
=  ^a-^byII,  1°; 

mp 

=  fl^  9  ,  by  definition; 

=  a'"'^  =  «"»". 
3°.  Let  n  be  any  rational  negative  quantity  and  =  —  jo. 

ThenK)-^=   _I_^J__^a-p. 

We  know  already  that  the  law  holds  when  m  and  7i  are 
positive  integers. 

Hence  for  all  rational  values  of  ?n  and  n 

III.  To  prove  (aby  =  «"Z>"  for  all  rational  values  of  7i. 
1°.  Let  n  be  any  positive  rational  quantity  which  may 

be  denoted  by  a  fraction -,  in  which  p  and  5'  are  positive 

integers. 

Then  (aby  =  (ah)^  =  ^{aby,  by  definition; 
=  ^a^h^,  by  (2). 


^04  THE  INDEX  LAW. 

Also,  {a'^lf'Y  for  all  values  of  7i 

=  fl^"^"  .  ft"Z»^  .  a'^h'' ioq  factors 

—  aJ^ .  oT' .  a"" .  .  ,io  q  factorsX  ^''  .&".&"... 
to  §*  factors 

.  •.     a^'lf  =  ^«"«^»««. 
But,  since     n  =  --  or  nq  =  p. 

Therefore  for  all  positive  rational  values  of  n 

(aby  =  a''b\ 

2°.  Let  n  be  any  rational  negative  quantity  and  =  —  p, 
p  being  a  positive  integer.     Then 

w  =  {"i}-' = p)-.  =  i= »-"  • "-"  =  «"*"• 

We  know  already  that  the  law  holds  when  m  and  n  are 
positive  integers. 

Hence  for  all  rational  values  of  n 

(ahy  =  a/'b\ 

EXERCISE  XCII. 
I. 

Find  the  values  of: 

1.      642/s.         2     16-8/2^  3     25-1/2. 

4.       Q-f.     5.     (100000)-/=       e.     (4)"^. 


THE  INDEX  LAW.  205 

Simplify : 

9.   {a-y%-^)-\  10.  {d'by'')-yK 

Express  with  fractional  or  negative  indices : 
11.      Va-\-  Vh  -\-  \^x^.         12.      V^^  +  ^^ay\ 


13.      4/«V  +  VaY-  14.      VxYz^  +  VaV. 

Express  without  fractional  or  negative  indices : 

16.  xV^  -  z-^  16.     a-^-y^ 

17,  ^3/4J-2  _  ^-3/4J2,  18         a-SJ-2/3  +  3^1/3^,-3/4. 

Multiply : 

I. 

19.  a;2/5  _^  y2/5  ^  ^2/5  _  y2/5^ 

20.  1  +  a;V5  -f  a;2/5  by  1  -  a;V5. 

21.  «V2  _  ^1/4^,1/4  _|_  Jl/2  by  «l/4  4.  ^1/4. 

II. 

22.  icVe  _  a;V6  _|_  :^;i/2  _  ^1/6  _|_  ^-1/6  _  ^-3/6   i^y    ^^/e  _|_ 

23.  x^  +  :z;3/2  +  1  by  ^"^  +  x-^/^  +  1. 


'^*-     ^^  -  3^^^'^^^'^  +  i«^'^^^'^  -  i'     ^^  r^'^ 


4 


Divide : 

I. 

25.     x^  —  y^  by  ^/^  —  y^  '^. 


206  THE  INDEX  LAW. 

6w  6n  2n  2n 

26.  x^  —y^  by  x'"*  —y^. 

27.  x^  +  y^  by  2;V3  .f  ^4/3_ 

28.  x^  +  32?/5/^  by  x^/^  +  2«/V4. 

II. 

29.  x''/^  -  2  +  ^-V3  by  .^2/3  -  x-y\ 

30.  «'/2  -  X  by  «Vio  _  ^4/5, 

81.     a;^/^  —  2;^^/^  +  xy^y  —  3/^/^  by  a;V2  _  yV^, 


CHAPTER    XVIII. 
ELIMINATION. 

160.  Simultaneous  and  Independent  Eq[uations. — Two 

or  more  equations  are  said  to  be  simultaneous  when  they 
are  satisfied  by  the  same  values  of  their  unknown  quanti- 
ties. 

The  equations  are  independent  when  one  cannot  be  de- 
rived from  the  other. 

When  an  equation  contains  two  or  more  unknown 
quantities,  an  indefinite  number  of  values  of  their  quanti- 
ties may  be  found  which  will  satisfy  the  equation. 

e.g.     Let  3x  +  4?/  =  18. 

Transpose  the  term  containing  y  and  solve  for  x,  and 
we  have 

^-  .     3       • 

If  in  this  result  we  put  ?/  =  3,  we  get 
18-12       ^ 

and  if  we  put  ^  =  4,  we  get 

18  -  16       2 

From  this  it  appears  that  when  an  equation  contains 
two  unknown  quantities,  it  can  be  satisfied  by  an  unlimited 
number  of  pairs  of  values  of  these  quantities,  for  by  assign- 

807 


208  ELIMINATION. 

ing  any  value  whatsoever  to  one  of  these  quantities  we  ob- 
tain an  equation  from  which  the  other  may  be  found. 
In  general  terms,  if 

ax -{- hy  -\-  c  =  0, 

we  may  give  y  any  value  m.     Then  ^^ 

ax  -\-  hm  -f-  c  =  0, 

hn  +  c 

x= — . 

a 

The  values  y  —  m,  and  x— ^^,  evidently  satisfy 

the  given  equation.  That  is,  in  an  equation  of  the  first 
degree  in  x  and  ?/,  to  every  value  of  y  there  is  a  correspond- 
ing value  of  X  which  will  satisfy  the  equation. 

161.  Two  Unknown  duantities  require  two  Inde- 
pendent Equations  for  their  Solution.  —  If,  however,  we 
have  two  independent  equations  in  x  and  y,  of  the  indefi- 
nite number  of  pairs  of  values  of  x  and  y  which  will  satisfy 
either  equation  alone,  there  is  only  one  pair  which  will 
satisfy  both. 

To  obtain  this  pair  of  values,  we  may  solve  each  equa- 
tion for  the  same  letter,  and  put  the  resulting  values  equal. 

e.g.  Let  3a:  +  4?/  =  18,  (1) 

and  2:c+5«/=:19.  (2) 

From  (1),  we  have  x  — — ^, 

o 

and  from  (2),  x  — — ^. 

Now  as  we  are  seeking  the  value  of  x,  which  is  the  same 
in  both  equations,  we  may  put 

18-4y_19-5y 


ELIMINATION.  209 

As  this  is  a  simple  equation  of  the  first  degree  in  y,  we 
may  solve  it  for  y,  and  then  find  the  value  of  y  which  will 
give  the  same  value  of  x  in  the  two  equations. 

Solving  (3)  for  y,  we  obtain  y  =  3. 

Substituting  this  value  of  y  in  (1),  we  get 

3:?;  +  12  =  18. 
.-.     3:^^  =  6, 
and  X  =  2. 

The  same  value  of  x  would  have  been  obtained  had  we 
substituted  the  value  of  y  in  (2). 

162.  Elimination. — The  general  method  of  solving  si- 
multaneous equations  of  two  or  more  unknown  quantities  is 
to  get  rid  one  after  another  of  all  the  unknown  quantities 
but  one,  so  as  to  obtain  an  equation  containing  that  un- 
known quantity  alone;  then  to  find  the  value  of  this 
quantity  from  the  resulting  equation,  and  afterwards  of  the 
remaining  unknown  quantities  by  substitution. 

The  process  of  getting  rid  of  the  unknown  quantities  is 
called  elimination. 

163.  Three  Methods  of  Elimination. — There  are  three 
general  methods  of  elimination,  known  respectively  as  the 
methods  by  comparison,  by  substitution,  and  by  addition  or 
suMradion. 

The  first  has  been  illustrated  already.  It  consists  in 
finding  the  value  of  the  same  unknown  quantity  from  each 
of  the  two  equations,  and  putting  their  values  equal  to  each 
other. 

e.g.  2a; +  3^  =  19;  (1) 

•    -  3x-\-'ly  =  16.  (2) 

From  (1),  X  =  — --^-, 


210  ELIMINATION. 

and  from  (2),  x  =  -— ^ . 

o 

19  -  3y  _  16  -  %y 
2        ~        3       ' 

or  57  -  9?/  =  32  -  4.y, 

or  5«/  =  25. 

.-.     y=    b. 

Substituting  this  value  of  ?/  in  (1),  we  get 

22; +  15  =  19; 

x  =  2. 

The  second  method  consists  in  finding  the  value  of  one 
of  the  unknown  quantities  from  one  of  the  equations,  and 
substituting  that  value  in  the  other. 

e.g.  2x-{-'dy  =  19;  (1) 

Sx  +  2y  =  16.  (2) 

From  (1),  we  obtain  x  = ^r — -. 

Substituting  this  value  for  ^  in  (2),  we  get 


"V^*^ 

2           ^^ 

y  = 

16, 

or 

57 

-9.V  +  4^: 

=  32, 

or 

by  =  25. 

*•    y  =  ^' 

Substitute  this  value  in  (1)  or 

{% 

and 

we 

find 

x=    2. 

The  third  consists  in  multiplying  each  of  the  equations 
by  some  number  which  will  make  the  coefiicients  of  one  of 


ELIMINATION.  2 1 1 

the  unknown  quantities  the  same  in  both,  and  adding  the 
equations  when  these  coefficients  have  opposite  signs  in  the 
two  equations,  and  subtracting  the  equations  when  the 
coefficients  have  the  same  signs  in  both. 

e.g.  2a: +  3?/ =  19,  (1) 

dx-\-^  =  16.  (2) 

Multiplying  the  first  equation  by  3  and  the  second  by  2, 
we  have  Qx  -{- 2y  =  57,  (3) 

and  6:r  +  4?/  =  32.  (4) 

Subtracting  (4)  from  (3),  we  get 
by  =  25. 
.-.      ^  =  5, 
and  X  =  2. 

The  third  method  is  the  one  usually  employed,  and  the 
first  is  least  used.  The  student  should,  however,  be  familiar 
with  the  use  of  all  three. 

EXERCISE  XCIII. 

Solve  the  following  equations  by  each  of  the  three 
methods : 


I. 

3:c  + 

4:X- 

y=    9, 
%y=    2. 

2. 

bx-    %y=    5, 
2a;  +      y  =  11. 

%x- 

%x- 

6?/  =  10, 
72/  =  ->  3. 

4. 

Ix  +  lly  =  17, 
2x  -    by  =  13. 

9x- 
3^  + 

5^  =  -  1, 
6?/  =  15. 

6. 

11a;  +    ly=-    5, 
4a;  -    by^-  32. 

%x-\- 

7x-\- 

2/=    4, 
8i/  =  -  13. 

8. 

8a;  -      1/  =  -    6, 
a;+    By  =-17. 

212  ELIMINATION. 

9.   14a;  —    dy  —  45,  lo.      6x  —    7y  —    0, 

6x  +  17^  =1.  7x-\-    5ij  =  74.      ' 

Solve  the  following  by  any  method  of  elimination : 
11.     -3-  +  2/  =  10,  12.     2x  -  ^—j-  =  4, 

,  y       ^  o       r^      ^  —  ^ 

^+1=    5.  3^  =  9--^. 

13.  Find  the  first  four  terms  of  the  square  root  of  1  —  a:. 

14.  Find  the  cube  root  of  cc^  —  ^x^  + 15^;*  —  20:^3  +  15x^ 
-6x-\-l. 


II. 
3x^1/       ,  „  2x  —Sy      Sx  —by       1 

^_X=s  :r  +  4y      5.T  -  4y  _ 

3         8  11      "^        7         - 

17.  -yC^  +  2/)  =    5(^  -  «/)^ 
—(a;  +  2/)  =  35-(^  -  «/)  -  y 

18.  :?:(2/  +  7)  ==  y{x  +  1), 

2:^;  =  3^/  -  19. 

4  5 

2  ^3~  -  ^^  2       • 

20.    «.'^  =  %,  21.    :^  +   y  =  h 

X  -\-  y  —  c.  ax  -\-  hy  —  c. 

22.     ic  +  ?/  =  a  +  J,  23.     a;  +?/  =  «  +  ^, 

X  -\-  a  _  h  ax  -\-  hy  =  a;^  -\-  h^. 

y  -\-h  ~"^' 


ELIMINATION.  213 

a  h  X  A-  a    ,    b 

ax+  by  =  c.  y  -\- b        a 

EXERCISE  XCIV. 

Solve  the  following  problems  by  two  unknown  quan- 
tities : 

Ex.  1.  Find  two  numbers  whose  sum  is  17  and 
whose  difference  is  3. 

Let  a;  =  the  larger  number, 

and  y  =  the  smaller  number. 

Then  x  +  y  =  15,  (1) 

and  X  -  y  =    3.  (2) 

Add  equation  (2)  to  equation  (1),  and  we  get 
2x  =  18. 
.-.     x=    9. 

Subtract  equation  (2)  from  equation  (1),  and  we  get 
2y  =  12. 
.-.     y=    Q. 

Hence  the  numbers  are  9  and  6. 

2.  Find  a  fraction  such  that  when  5  is  added  to  its 
numerator  and  2  is  added  to  its  denominator,  its  value  is 
3/4;  and  if  1  be  subtracted  from  its  numerator  and  5  be 
subtracted  from  its  denominator,  its  value  is  3/5. 

Let  X  =  the  numerator, 

and  y  =  the  denominator. 

Then  — {—  =  -, 

y^2       4 


214  ELIMINATION. 

X  -  1       3 

and  k~'k- 

y  -  b       5 

Clearing  of  fractions,  we  have 
4a:  +  20  =  3«/  +    6,     or    4:X  -  dy  =  -  14,    (1) 
and 

6x  -    6  =  3y  -  15,     or     5x  -  Sy  =  -  10.    (2) 
Subtracting  (1)  from  (2),  we  get 

X    =    4:. 

.'.    16  -  3^  =  -  14, 

or  dy  =  30. 

.-.     y  =  10. 

Hence  the  fraction  is  4/10. 

3.  There  is  a  number  composed  of  two  digits.  The 
sum  of  the  digits  is  7,  and  if  9  be  added  to  the  number  the 
digits  will  be  reversed. 

Let  X  =  digit  in  the  tens^  place, 

and  y  =  digit  in  the  units'  place. 

Then  the  number  is  10a:  +  y.  When  the  digits  are 
reversed  the  number  is        lOy  -\-  x. 

Then  x -\- y  =  7,      ^  (1) 

10x  +  y  +  9  =  10y  +  x, 
or  9a;  —  9i/  =  —  9, 

or  X  —  y  =  —  1.  (2) 

Adding  (1)  and  (2),  we  get 

%x  =  6. 
.\     X  =  3, 


ELIMINATION.  215 

Subtracting  (2)  from  (1),  we  get 
2^  =  8. 
.-.      y  =  ^> 
Hence  the  number  is  34. 
I. 

1.  The  sum  of  two  numbers  is  8  and  their  difference  is 
G.     What  are  the  numbers  ? 

2.  There  is  a  certain  fraction,  such  that  if  its  numer- 
ator be  increased  by  4,  its  value  is  4/5 ;  and  if  its  denom- 
inator be  increased  by  one,  its  value  is  1/2.  What  is  the 
fraction  ? 

3.  A  certain  number  of  two  digits  is  equal  to  five  times 
the  sum  of  its  digits,  and  if  9  be  added  to  the  number,  its 
digits  will  be  reversed. 

4.  A  number  consists  of  two  digits  whose  difference  is 
1 ;  if  it  be  diminished  by  the  sum  of  its  digits,  the  digits 
will  be  reversed.     What  is  the  number  ? 

5.  Eight  years  ago  A  was  five  times  as  old  as  B,  and  in 
two  years  he  will  be  three  times  as  old.  What  are  their 
present  ages  ? 

6.  A  alone  does  3/5  of  a  piece  of  work  in  30  days,  and 
then  with  B's  help  finishes  it  in  10  days.  In  what  time 
could  each  do  it  alone  ? 

II. 

7.  A  man  buys  8  lbs.  of  tea  and  5  lbs.  of  sugar  for 
$2.39 ;  and  at  another  time  5  lbs.  of  tea  and  8  lbs.  of  sugar 
for  $1.64,  the  price  being  the  same  as  before.  What  were 
the  prices  ? 

8.  Two  vessels  contain  mixtures  of  wine  and  water.  In 
the  first  there  are  three  times  as  much  wine  as  water,  and  in 
the  second  five  times  as  much  water  as  wine.  How  many 
gallons  must  be  drawn  from  each  vessel  to  fill  a  third,  which 


216  ELIMINATION. 

holds  7  gallons,  with  a  mixture  which  shall  be  half  wine 
and  half  water  ? 

9.  Two  vessels  contain  mixtures  of  wine  and  water.  In 
the  first  there  are  4  gallons  of  wine  to  3  gallons  of  water, 
and  in  the  second  there  are  5  gallons  of  water  to  2  gallons 
of  wine.  How  many  gallons  must  be  drawn  from  each  ves- 
sel to  fill  a  third,  which  holds  12  gallons,  with  a  mixture 
which  shall  be  1/3  wine  ? 

10.  A  man  buys  2  lbs.  of  tea  and  6  lbs.  of  sugar  for  81 
cents,  and  at  another  time  4  lbs.  of  tea  and  9  lbs.  of  sugar 
for  $1.51|,  the  price  being  the  same  as  before.  What  were 
the  prices  ? 

164.  To  Solve  for  n  Unknown  Quantities  requires  7i  In- 
dependent Equations. — We  have  seen  that  we  need  two  si- 
multaneous equations  in  order  to  find  the  value  of  two  un- 
known quantities.  Similarly,  we  need  three  independent 
simultaneous  equations  in  order  to  find  the  value  of  three 
unknown  quantities,  and  n  independent  simultaneous 
equations  in  order  to  find  the  value  of  7i  unknown  quanti- 
ties. 

With  three  unknown  quantities,  we  first  combine  any 
pair  of  the  three  equations  so  as  to  eliminate  one  of  the  un- 
known quantities,  and  then  another  pair  so  as  to  eliminate 
the  same  unknown  quantity.  We  shall  then  have  two 
equations  with  two  unknown  quantities.  Then  we  combine 
these  two  equations  so  as  to  eliminate  one  of  the  remaining 
unknown  quantities,  and  thus  obtain  one  equation  with  a 
single  unknown  quantity.  From  this  we  obtain  the  value 
of  this  quantity,  and  then,  by  successive  substitution,  the 
values  of  the  other  two. 

e.g.  Qx-{-%y  -bz  =  13,  (1) 

^x-\-^  -2z  =  13,  (2) 

'7x-\-by  -^z  =  26.  (3) 


ELIMINATION.  217 

Eliminate  y  from  (1)  and  (2)  by  subtraction,  multiply- 
ing (1)  by  3  and  (2)  by  2. 

18a;  -\-Qy  -  15z  =  39, 
6x-i-6t/  -  4.z  =  26. 
.-.     12a; -11^  =  13.  (4) 

Next  eliminate  y  from  (1)  and  (3)  by  subtraction,  mul- 
tiplying (1)  by  5  and  (3)  by  2. 

30a;  +  10?/  -  26z  =  65, 
14a;  +  Wy  -    6z  =  52, 

.-.     16a; -19^  =  13.  (5) 

Next  eliminate  x  from  (4)  and  (5)  by  subtraction,  mul- 
tiplying (4)  by  4  and  (5)  by  3. 

48a;  -  44^;  =  52, 

48a;  -  57z  =  39. 

.-.     132=13, 

and  z  =  1. 

Remember  that  the  equations  may  be  combined  in  any 
order,  and  that  those  combinations  are  best  which  will  pro- 
duce the  required  result  in  the  simplest  and  most  direct 
way. 

EXERCISE  XCV. 


x-Jr2y-^2z  =  16, 
2a; -f    1/+    z  =  n, 
3a; +  42/+    ^  =  22. 

2.       a;  +  3^/  +  4^  =  7, 

a; +  2^+    z  =  0, 

2a;  +    2/ +  2^  =  6. 

X  +  4.y  +  3^  =  14, 
3a; +  3^+    2  =  21, 
2a;  +  22/+    z  =  Id, 

4.     3x-2y-\-    z  =  10, 
2a;  +  3«/  +    z  =  18, 

218                                 t:LIMINATI0n. 

5.     Sx-^4:y  =  0, 

6.      5:?;  +  2^  =  8f , 

2y  —  4:z=  —  14, 

Sz-y  =  1|, 

x-i-dy-\-2z=- 

1.               8a;  -  lOz  =  3f. 

II. 

7.     .-1  =  12, 

y  +  z      z  +  x      x-\-y 
^        5      -      4      -      3     ' 

^-1=14, 

X  +  y-\-Z  =  18. 

.-1=15. 

9.  %  =%  =5.- 

-3x,  10.   a;  +  16  =  ^  +  14 

2^  +  2  =  32;  -  3. 

=  3^  +  9 

n  n  ''[^  n  n 

11.  Multiply  3rr2  +  ^^^  —  5^*  ^J  ^^  —  2a;*. 

6n  6to,  n  m 

12.  Divide  a;  2  —  a;^  by  x^  —  x^, 

13.  Square  2a;V3  _  3^2/3  _j_  4^^ 

Note.  —  When  tliere  are  more  than  three  unknown 
quantities,  the  process  of  elimination  is  similar. 

EXERCISE  XCVI. 

Work  the  following  examples  by  three  unknown  quan- 
tities : 

I. 

1.  The  sums  of  three  numbers,  taken  two  by  two,  are 
20,  29,  and  27.     What  are  tlie  numbers  ? 

2.  The  sum  of  three  numbers  is  78,  1/3  the  difference 
of  the  first  and  second  is  4,  and  1/3  the  difference  of  the 
first  and  third  is  7.     What  are  the  numbers  ? 

3.  A  person  bought  three  silver  watches.    The  price  of 


ELIMINATION.  219 

the  first,  with  1/3  the  price  of  the  other  two,  was  40  dol- 
lars, the  price  of  the  second,  with  1/4  the  price  of  the  other 
two,  was  42  dollars,  and  the  price  of  the  third,  with  1/2 
the  price  of  the  other  two,  was  44  dollars.  What  was  the 
price  of  each  watch  ? 

4.  A,  B,  and  C  together  have  $2100.  Were  B  to  give 
A  300  dollars,  A  would  have  380  dollars  more  than  B,  and 
if  B  received  200  dollars  from  C,  they  would  both  have  the 
same  sum.     How  many  dollars  has  each  ? 

5.  A,  B,  and  C  can  perform  a  piece  of  work  in  20 
days,  A  and  B  in  30  days,  and  B  and  0  in  40  days.  How 
long  would  it  take  each  to  do  it  alone  ? 

6.  A  and  B  together  can  do  a  piece  of  work  in  6  days, 
B  and  0  in  6f  days,  and  A  and  0  in  b^^  days.  How  long 
would  it  take  each  to  do  it  alone  ? 

7.  A  number  is  composed  of  three  digits  whose  sum  is 

9.  The  digit  in  the  units'  place  is  twice  the  digit  in  the 
hundreds'  place,  and  if  198  be  added  to  the  number,  the 
digits  will  be  reversed.     What  is  the  number  ? 

8.  A  number  is  composed  of  three  digits  whose  sum  is 

10.  The  middle  digit  is  equal  to  the  sum  of  the  other 
two,  and  if  99  be  added  to  the  number  its  digits  will  be 
reversed.     What  is  the  number  ? 

9.  A  number  is  composed  of  three  digits  whose  sum  is 
14.  Seven  times  the  second  digit  exceeds  the  sum  of  the 
other  two  by  2,  and  if  the  first  and  second  digit  be  inter- 
changed the  resulting  number  will  be  less  than  the  given 
number  by  180.     What  is  the  number? 

II. 

10.  A  and  B  can  do  a  piece  of  work  in  r  days;  B  and 
C  in  s  days;  and  A  and  C  in  /  days.  In  how  many  days 
can  each  do  it  alone  ? 


/ 


220  ELIMINATION. 

Do  the  following  by  two  unknown  quantities: 

24.  A  crew  can  row  10  miles  in  50  minutes  down 
stream  and  12  miles  in  an  hour  and  a  half  up  stream. 
What  is  the  rate  in  miles  per  hour  of  the  stream,  and  of 
the  crew  in  still  water  ? 

Let  X  —  the  rate  in  miles  per  hour  of  the  crew  in  still 
water, 
and       y  =  the  rate  in  miles  per  hour  of  the  current. 

.*.   X  -\-  y  =  the  rate  in  miles  per  hour  of  the  crew 
down  stream, 
and       X  —  y  =  the  rate  in  miles  per  hour  of  the  crew  up 
stream. 

Since  the  number  of  miles  rowed,  divided  by  the  rate 
in  miles  per  hour,  is  equal  to  the  time  in  hours,  we  have 

10  5 


x-^y       6' 

and 

12          3 

x-y-  2' 

.*.     X  =  10,     and     ?/  =  2. 

25  A  crew  can  row  20  miles  down  stream  in  an  hour 
and  20  minutes,  and  18  miles  up  stream  in  2  hours.  What 
is  the  rate  of  the  current  in  miles  per  hour,  and  what  is 
the  rate  of  the  crew  in  still  water  ? 

26.  Two  trains  start  from  two  stations  at  the  same 
time,  and  each  proceeds  at  a  uniform  rate  towards  the 
other  station.  They  meet  in  twelve  hours,  and  one  has 
gone  108  miles  farther  than  the  other,  and  then  if  they 
continue  to  travel  at  the  same  rate  they  will  finish  their 
journey  in  9  hours  and  16  hours  respectively.  What  is 
the  rate  of  the  trains,  and  tlie  distance  between  the  towns  ? 


ELIMINATION.  221 

27.  Two  trains  start  from  two  stations  at  the  same 
time,  and  each  proceeds  at  a  uniform  rate  towards  the 
other  station.  They  meet  in  six  hours,  and  one  has  gone 
30  miles  farther  than  the  other,  and  then  if  they  con- 
tinue to  travel  at  the  same  rate,  they  will  finish  the 
journey  in  7  hours  and  12  minutes,  and  in  5  hours,  respec- 
tively. What  is  the  rate  of  the  trains,  and  what  is  the 
distance  between  the  towns  ? 

28.  A  certain  number  of  persons  paid  a  bill.  Had 
there  been  10  more,  each  would  have  paid  $2  less,  and 
had  there  been  5  less,  each  would  have  paid  $2.50  more. 
How  many  were  there,  and  how  much  did  each  pay  ? 

29.  A  sum  of  money  is  divided  equally  between  a  cer- 
tain number  of  persons.  Had  there  been  m  more,  each 
would  have  received  a  dollars  less ;  if  n  less,  each  would 
have  received  h  dollars  more.  How  many  persons  were 
there,  and  how  much  did  each  receive  ? 


CHAPTEK  XIX. 
QUADRATIC  EQUATIONS. 

A.       SUED   AND   IMAGINAEY   FACTORS. 

165.  Trinomial  and  Binomial  Quadratics. — A  complete 
quadratic  exj^ression  in  one  unknown  quantity  contains 
three  terms,  one  containing  the  square  of  the  unknown 
quantity,  one  containing  the  first  power  of  the  unknown 
quantity,  and  the  third  without  the  unknown  quantity. 
The  most  general  form  of  such  an  expression  is 

ax^  -\-  bx  -\-  c. 

The  term  which  does  not  contain  the  unknown  quan- 
tity is  called  the  constant  term  of  the  expression,  and  the 
complete  expression  is  called  a  trinomial  quadratic. 

When  the  term  containing  the  first  power  of  the 
unknown  quantity  is  wanting,  the  expression  becomes  a 
binomial,  and  is  called  an  incomplete  or  a  binomial 
quadratic  expression. 

166.  Factors  of  x^  +  c. — Every  binomial  quadratic  of 
the  form 

x^  -\-  c 

may  be  factored  as  the  difference  of  two  squares,  since  it 
may  be  written  in  the  form 

x^  -  {-  c). 


SURD  AND  IMAGINARY  FACTORS.  223 

The  factors  will  be 


X  -{-  V  —  c  and  x  —  V  —  c. 

1°.  When  c  represents  a  positive  number,  these  factors 
are  imaginary. 

2°.  When  c  represents  a  negative  number  which  is  not 
a  perfect  square,  the  factors  are  surd. 

3°.  When  c  represents  a  negative  number  which  is  a 
perfect  square,  the  factors  are  rational. 

e.g.      1.  x^^b=x^-(-b)  =  (x  -  V'^){x^  V^), 
a;2+4=^-2_(_4)  =  (a;  -  V^{x  +  V-l) 

=  {;x  -  2  i/^)(:^:4-2  V^. 

2.  a;2  +  (-  3)  =  3-2  -  3  =  {x  -  V^){:x  +  V^). 

3.  x^  +  (-  9)  =  a:2  -  9  =  (a;  -  Z){x  +  3). 

When  the  expression  is  in  the  form 

ax^  -\-  c, 

a  may  be  taken  out  as  a  factor  first,  and  then  the  remain- 
ing factor  may  be  factored  as  the  difference  of  two  squares. 
Thus, 

ax^  +  o  =  a[x^  +  '-)=a(^x^-[-'-]^ 

e.g.       1°.  3r?;2  +  6  =  3(a;2  +  2)  =  'd{x^  -  (-  2)) 

=3  3(:r  -  V^^2)(a;  +  V^^^). 
r.  ix^^{-^0)  =  4.(x'-b) 

=  4:(x  -f  V5)(x  -  Vb). 


224  QUADRATIC  EQUATIONS. 

3°.  6a;2+(-20)  =  5(2;2-4) 

=  b{x  -  2)(^  +  2). 

=  ^x  -  VbJ^){x  +  1/573). 
4a;2  4.  (_  3)  :=  4(a;  _  |/374)(2;  +  VsTT) 

=4-irt){.+i^). 

EXERCISE  XCVII. 

Factor  the  following  quadratic  expressions : 

1.     a;2  +  5.  2.     x^  -  7.  3.  a;^  +  16. 

4.     Zx^  -  9.  5.     5a;2  -  25.  6.  '^x^  +  14. 

7.     2a;2  _  3,  8      3^2  _|_  5^  9  5^2  _  2. 

10.     4a;2  +  3.  11.     dx^  -  4.  12.     Ix^  +  5. 

167.  Factors  of  a  Trinomial  Cluadratic.  —  Every  tri- 
nomial quadratic  expression  may  be  factored  as  the  differ- 
ence of  two  squares. 

We  first  take  out  the  coefficient  of  the  square  of  the 
unknown  quantity,  and  after  the  second  term  of  the  ex- 
pression we  add  and  subtract  the  square  of  half  the  coef- 
ficient of  the  first  power  of  the  unknown  quantity.  This 
will  give  a  polynomial  of  five  terms,  the  first  three  of  which 
will  be  a  perfect  square.  The  last  two  terms  must  be  com- 
bined into  one  with  a  minus  sign  before  it.  The  factors 
will  both  be  real  when  this  term  is  essentially  positive, 
rational  when  it  is  an  exact  square,  and  surd  when  it  is  not 


SURD  AND  IMAGINARY  FACTORS.  225 

an  exact  square.     The  factors  will  both  be  imaginary  when 
this  last  term  is  essentially  negative. 

e.g.     1°.  Factor     ^x^  +  15a;  +  18. 

First,  we  have         Zx^  +  15:^;  +  18  =  ^x^  +  6x -\-  6). 

Then,  after  the  second  term  of  the  second  factor,  add 
and  subtract  (5/2)^,  and  we  get 


('+l+l)('+l-8 


.  •.     3a;2  +  15^;  +  18  =  ^x  +  3)(a;  +  2). 
2°.  Factor  ax^-\-bx^c. 

First,  ax^  -{-  hx  -{-  c  =  aix^  A — x  A — ). 
\  a        al 

Then    ^  +  ^  +  ^=-'  +  ^  +  £-£  +  „- 

=  x^  -^-x^  —  -  ^'  ~  ^^^ 
a         4«^  4a^ 

h    ,    Vl>^  -  4:ac\f     ,     b        Vb^-  4.ac' 


=  ("+2^+ 2^— jl^  +  2^- 

/     ,    b  +  Vb^-  4:ac\/     , 


2« 


b-  Vb^-  4ac 


2a 
ax^  -\-bx  -\-  c 


b^-S/W  -  4.ac\f     ,   b-  Vb^  -  4:ac\ 


=  T  + 2a k  + 


2a 


Whether  these  factors  be  rational,  surd,  or  imaginary 
depends  upon  the  radical  Vb'^  —  4ac, 


226  QUADRATIC  EQUATIONS. 

If  the  quantity  under  the  radical  be  positive,  the  factors 
will  be  real. 

If  also  the  quantity  under  the  radical  be  a  perfect 
square,  the  factors  will  be  rational;  and  if  this  quantity  be 
not  a  perfect  square,  the  factors  will  be  surd. 

If  the  quantity  under  the  radical  be  0,  the  factors  will 
be  equal. 

If  the  quantity  under  the  radical  sign  be  negative,  the 
factors  will  be  imaginary. 

Since  ax^  -\-'bx-\-  c  is  the  general  form  of  a  trinomial 
.quadratic  expression, 


I     ,   ^  +  y^^  ^(^c\(     ,   h  -  Vh^-  4:ac\ 

may  serve  as  a  formula  by  which  all  such  expressions  may 
be  factored. 

e.g.  Factor  ^x^  -f-  4a:  +  5. 

Comparing  this  with  ax^  -\-bx-\-  c,  we  see  that  a  =  3, 
b  =  4:,  and  c  =  5. 

Substituting  these  values  in  the  formula,  we  get 


4  +  VU  -  60\/      ,    4  -  Vl6  -  60^ 


^^+— — ^ — U^+ 


or 


or 


In  this  case  the  binomial  factors  are  imaginary. 

EXERCISE  XCVIII. 

Factor  the  following   trinomial   quadratic   expressions 
by  the  formula: 

I. 

1.     4:X^-\-7x-6,  8,     2x^-]-6x  +  2, 


ROOTS  OF  AN  EQUATION.  227 

3.     6x^  —  ^x  —  7.  4.     'ox^  —    ^x  —  '6. 

5.     ^x'^'dx-\-Q.  6.     2:^2  + 10a; +  8. 

7.  A  man  bought  175  acres  of  land  for  6000  dollars. 
For  a  part  of  it  he  paid  40  dollars  an  acre,  and  for  the 
remainder  25  dollars  an  acre.  How  many  acres  in  each 
part  ? 

II. 
8.     7;?;2  +  9a;  +  2.  9.    lx^-\-'^%x-    7. 

10.     ^x^  +  7a;  -  6.  11.    4^2  -  Mx  +  12. 

12.  Ux^  -\-    x-Q>.  13.    3a;2  -  10.C  +    6. 

14.  A  man  bought  m  acres  of  land  for  s  dollars.  For  a 
part  of  it  he  paid  a  dollars  an  acre,  and  for  the  remainder 
1)  dollars  an  acre.    How  many  acres  were  there  in  each  part  ? 

V4a;  +  1  +  2  Vx        ^ 
15    Solve  — =z=r —  =  9. 

V4:x  +  1  -  2  Va; 


Vx  -\-  a  4-  Vx 
16.    Solve  — ^=^= — :  =  c. 

Vx  4-  a  —  Vx 


B.       ROOTS   OF   AInT    EQUATIOI^. 

168.  Quadratic  Equations. — A  quadratic  equation  of 
one  unknown  quantity  is  an  equation  whose  first  member 
is  a  complete  or  an  incomplete  quadratic  expression  in  that 
letter  after  the  equation  has  been  reduced  to  its  simplest 
form  and  all  its  terms  have  been  transposed  into  its  first 
member.  After  reduction  and  transposition  the  equation 
takes  either  the  form 

ax^  -{-hx  -^  c  =  0  (1) 

or  ax^  4-  c  ==  0.        .  (2) 

169.  Boots  of  an  Equation. — A  root  of  an  equation  is  a 
value   of  its   unknown   quantity   which   reduces  its  first 


228  qUADBATIG  EQUATIONS. 

member  to  zero,  after  it  has  been  reduced  to  the  form  of 
(1)  or  (2). 

170.  Solution  of  a  Quadratic  Equation. — To  solve  a 
quadratic  equation  is  to  find  its  roots,  or  the  values  of  its 
unknown  quantity  which  will  reduce  to  zero  the  first 
member  of  the  equation  after  it  has  been  brought  into  its 
type  form. 

Since  a  product  is  zero  when  any  one  of  its  factors  is 
zero,  the  values  of  its  unknown  quantity  which  will  reduce 
to  zero  the  factors  of  the  first  member  after  it  has  been 
brought  into  its  type  form  are  the  roots  of  the  equation. 
Hence,  to  solve  a  quadratic  equation,  reduce  it  to  the  type 
form,  factor  its  first  member,  equate  each  factor  to  zero, 
and  solve  for  its  unknown  quantity. 

e.g.     Solve       a:^  —  6x  =  —  8. 

Reduced  to  the  type  form  this  becomes 
x^  -  Qx  -\-  S  =  0, 
or  (x  -  ^)(x  -  4)  =  0. 

Put  X  -%  =  0, 

and  we  have  a;  =  2. 

Put  X-  4  =  0, 

and  we  have  a;  =  4. 

Hence  2  and  4  are  the  roots  of  the  equation,  for  either 
of  these  values  of  x  will  reduce  the  first  member  of  the  type 
form  to  zero. 

We  have  seen  that  every  quadratic  expression  in  one 
letter  may  be  resolved  into  two  factors  of  the  first  degree 
in  that  letter.  Hence  every  quadratic  equation  has  two 
roots.  Moreover  a  product  cannot  vanish  unless  one  of  its 
factors  vanishes.  Therefore  a  quadratic  equation  has  only 
two  roots.  These  roots  will  be  rational  when  the  factors 
of  the  first  member  of  the  reduced  form  are  rational,  and 


ROOTS  OF  AN  EQUATION.  229 

equal  when  the  factors  are  identical;  surd  when  the  factors 
are  surds;  and  imaginary  when  the  factors  are  imaginary. 

e.g.     1.  Solve    x^  —  Qx  =  —  ^. 
When  reduced  to  the  type  form  this  becomes 
a:2  _  6^  +  9  =  0. 
.*.     {x-d){x-'d)=Q. 

Therefore  the  roots  are  3  and  3,  and  are  rational  and 
equal. 

The  roots  of  a  quadratic  equation  are  equal  when  the 
first  member  of  the  reduced  form  is  a  perfect  square. 

2.  Solve  x'  -  lla:  =  -  28. 
Transposing,  we  have 

x^  -  11a;  +  28  = 
.-.     {x-^){x-l)  =  0. 
.'.     a:  =  4  or  7. 

Therefore  the  roots  of  the  equation  are  4  and  7,  and  are 
rational  and  unequal. 

3.  Solve  x^-4:x  +  l  =  0. 

Bring  the  first  member  of  this  equation  under  the  case 
of  the  diiference  of  two  squares  by  adding  and  subtracting 
the  square  of  half  the  coefficient  of  x,  and  we  have 

iK2  -  4x  +  4  -  3  -  0. 

.-.     (x-2-\-  V3)(x  -2  -  V3)  =  0. 

,',      a:  =  2  -  V3    and  2  -f  V3. 

Therefore  the .  roots  of  the  equation  are  2  —  V3  and 
2  +  V'd,  and  are  surd  and  unequal. 

4.  Solve  a;2  -  6:r  +  11  =  0. 


230  QUADRATIC  EQUATIONS. 

Bring  the  first  member  under  the  case  of  the  difference 
of  two  squares  by  adding  and  subtracting  the  square  of 
half  the  coefficient  of  x,  and  we  have 

a;2- 6:?;  +  9  -  (- 2)  =  0. 


(a;  -  3  +  y  -  2)(«  -'d-  V  ~  2)=0. 


.  •.     a;  =  3  —  |/  —  2   and   3  +  V  —  2. 
Therefore  tlie  roots  of  the  equation  are  3  —  i^  —  2  and 


3  4-  'Z  —  2,  and  are  imaginary. 

EXERCISE  XCIX. 

Solve  the  following  quadratic  ( 

3quations  by  factoring: 

1.     a;2  -  32;  -  18  =  0. 

X. 

2. 

x^  -\-  4:X  =  45. 

3.     x^  +  13a;  +  25  =  - 

15.     4. 

a;2  -  12a:  -  5  ==  -  40. 

6.     a;2  4-  4x  +  20  =  4  - 

-  4a;.    6. 

x^  —  5x  =  5x  —  25. 

7.     a;2  -  3  =  6. 

8. 
II. 

a;2  -  2«2   ::.:    _   «2. 

9.     x^-{-{a-^'b)x-\-al)^0. 

10. 

a;2+(«-J)a;-«J=0. 

11.     2a;2  4-  a;  -  3  =  0. 

12. 

3a;2  +  5a:  =  12. 

13.     15a;2  +  14a;  =  8. 

14. 

7a;2  +  15.T  =  -  8. 

15.     12  +  2a;2  ==  11a;. 

16. 

-  3x^  +  17a;  =  20. 

171.  Formation  of  Quadratic  Equations. — Since  we  ob- 
tain the  roots  of  a  quadratic  equation  by  equating  to  zero 
each  factor  of  the  first  member  of  its  type  form,  it  follows 
that  these  factors  are  the  unknown  quantity  of  the  equation 
minus  each  of  its  roots  in  turn. 

Hence  we  may  obtain  a  quadratic  equation  in  x  whose 
roots  shall  have  given  values  by  using  as  factors  x  minus 


HOOTS  OF  AN  EQUATION.  ^31 

each  of  the  given  roots  in  turn,  finding  the  product  of  these 
factors,  and  equating  this  product  to  zero. 

e.g.  1.  Form  the  quadratic  equation  in  x  whose  roots 
are  4  and  —  7.  The  factors  of  the  first  member  of  its 
type  form  will  be 

{x  —  4)     and     {x  +  7). 
...     (^_4)(.'^  +  7)  =  0, 
or  a;2  +  3a;-.28  =  0, 

which  is  the  required  equation. 

2.  Form  the  quadratic  equation  in  x  whose  roots  are 

3  +  V5  and  (3  -  Vb). 
Here  the  factors  are  x  —  (3  +  i^5)  and  a:  —  (3  —  V6). 
...     (2;  -  (3  +  V^)){x  -  (3  -  1/5))  =  0, 
or  a;2  -  6a;  +  4  =  0. 

EXERCISE  C. 

Form  the  quadratic  equations  in  x  whose  roots  have  the 
following  values : 

I. 

1.     3  and  7.         2.    4  and  —  6.  3.-7  and  —  1. 

4.     0  and  2.         5.-9  and  0.  6.     7  and  —  7. 

7.     —8  and— 8.  8.     11  and  11.  9.     3  and  3/4. 

•3  +  4/7        ,3-1/7  7+4/^5'      ,7-4/5 

,0.     -^_  and  — ^.     11.  -^-  and  —-. 


12.  4  +  4/-  6  and  4  -  4/-  6. 

II. 

13.  -  2/3  and  -  5/6.     u.     3/2  and  -  1. 

15.     7  and  -  2/5.  16.     3  +  4/5  and  3  -  1^, 


232  QUADRATIC  EQUATIONS. 

17.     2  +  l/8~and  2-  l/8.       is.     5  +  ^3  and  5  -  l/3. 


19.  9  +  y-  4  and  9  -  y-  4. 

7  +  1/3-3  7  _  |/3^ 

20.  _^^—  and— ^^ . 

„,     11  +  4/"=^      ,  11  -  V^l 

21.     1^—  and j^_. 

I. 

22.  Reduce  —  — ^  -| —  to  a  single  negative  fraction. 

23.  Reduce , — ~  —  a;  to  a  single  fraction. 

ic  +  2  ^ 

22;2   ^x 

24.  Reduce  2x — -  to  a  sinde  fraction. 


EXERCISE    CI. 


1.      (a:  -  2)2  -  1  =  |(:r  +  2). 


2.     2.^2  _^  2(a;  +  1)2  =:  :^2r(a;  +  1). 


13 
3 

3.  (2  -  xf  -  (2  -  x)(x  -  3)  +  (a^  -  3)2  =  1. 

4.  ^  +  i  =  4i.  5.     ^^+-^  =  2i. 

a;  +  2      2;  +  l  _  26  4  3      _  17 

®'     a:  +  l  +  «  H-  2  ~  y'         "'^     ic-  3  ~  a;  +  5  ~  10* 

II. 


HOOTS  OF  AN  EQUATION.  233 

4.x -d      .,   ,   2a;  -  3  x-1      x-\-l  6x 

10.   w- ^  =  3  + -.       11.  — — :  + 


3a:  -  7  'x-1'       "  '  x+1^  x-1       x^  -1' 

^x-1       o^  +  1  _  «  2x-l      13  _  3a;  +  5 


EXERCISE  Cll. 

1.     Solve  ^{3  -  4:x)  +  4/(2  +  5x)  =  ^(5  +  x). 
Transposing,  we  have 

|/(3  -  4.x)  +  |/(2  +  5a;)  -  |/(5  +  x)  =  0. 
Multiplying  by  the  conjugate,  we  have 
3  _  4^  _}_  2  |/(3  -  4a;)  V(2  +  5a;)  +  2  +  5a;  -  5  -  a;  =  0, 
or  2  |/(3  -  4)  4/(2  +  5a;)  =  0. 

.-.      |/(3  -  4a;)  |/(2  +  5a;)  =  0. 
.-.     (3  -  4a;)(2  +  5a;)  =  0. 
.-.     a;  =  3/4  and  -  2/5. 


2.  i/(5  -  7a;)  +  |/(4a;  -  3)  =  |/(2  -  3a;). 

3.  Vi^  +  «)  +  V(^  -^)  =  V(^^  -{-a-b). 

4.  |/(3  +  4a;)  -  i/(4  +  2a;)  =  |/(7  +  6a;). 
a.  4/(2  -  3a;)  -  |/(7  +  a;)  :=.  4/(5  +  4.x), 

6.  V(a;2H-3a;-54)-  4/(a;2-3a;-54)=  |/(2a;2  -108). 

II. 

7.  4/(a;2+4a;-60)-  |/(a;2-4a;-60)=  ^{2x^-120). 

8  |/(12a;2-a;-6)-  |/(12a;2  +  a;-6)=  i/(24a;2-12). 

9.  4/(36a;2+24a;+l)  +  V(36:i;2-24a;+l)  =  t/(72a;2+2). 


234  QUADRATIC  EQUATIONS. 

172.  Interpretation  of  Solutions. — 

Ex.  1.  A  man  sold  a  watch  for  24  dollars  and  lost  as 
many  per  cent  as  there  were  dollars  in  the  cost  of  the  watch. 
What  was  the  cost  of  the  watch  ? 


Let 

X  =  the  cost  in  dollars. 

Then 

X  =  the  lost  per  cent, 

and 

^•ioo=ioo=i°'''°'i°"«'^- 

Also, 

X  —  24t  =  loss  in  dollars. 

••• 

100=^     '''■ 

Solving 

this 

,  we  get 
x=QO  or   40. 

That  is,  the  cost  was  either  60  dollars  or  40  dollars ;  for 
either  of  these  values  satisfies  the  conditions  of  the  problem. 

2.  A  farmer  bought  a  number  of  sheep  for  80  dollars. 
Had  he  bought  4  less  for  the  same  money,  they  would  have 
cost  him  1  dollar  apiece  more.    How  many  did  he  buy  ? 

Let  X  =  the  number  bought. 

80 
Then       —  =  the  price  per  head  in  dollars, 

80 
and  J  =  the  price  per  head,  if  there  had  been  4 

more. 

80         80 


X  X   —   4: 


- 1. 


Solving  this  equation,  we  get  a;  =  —  16  or  +  20. 
Only  the  positive  value  will  satisfy  the  condition  of  the 
problem.     Therefore  the  number  of  sheep  was  20. 

In  solving  problems  which  involve  quadratics,  tliere 


ROOTS  OF  AN  EQUATION.  235 

will  be,  in  general,  two  values  of  the  unknown  quantity, 
both  of  which  may  not  answer  to  the  conditions  of  the 
problem.  This  is  due  to  the  fact  that  the  symbolic  lan- 
guage of  algebra  is  more  general  than  ordinary  language. 
So  that  the  equations  which  correctly  represent  the  con- 
ditions of  the  oral  problems  may  represent  other  allied 
conditions  also.  The  equation  is  entirely  general,  while 
the  verbal  statement  is  more  or  less  restricted.  Verbal 
statements  are  supposed  generally  to  be  restricted  to  an 
arithmetical  sense  which  admits  only  of  positive  numbers ; 
while  there  is  no  restriction  on  the  numerical  symbols  of 
an  algebraic  equation. 

A  little  consideration  will  enable  the  pupil  to  determine 
whether  or  not  both  values  of  the  unknown  quantity  will 
fit  the  conditions  of  the  verbal  problem,  and  which  one  to 
select  in  case  both  will  not  answer.  It  will  be  found  also 
a  valuable  exercise  to  interpret  negative  results  when 
possible. 

Thus  in  the  last  example,  to  buy  —  16  sheep  has  no 
meaning  in  the  arithmetical  sense,  but  algebraically  it 
means  to  sell  16  sheep. 

To  buy  4  less  than  —  16  would  mean  to  sell  20. 

In  the  first  case  he  would  have  paid  —  $5  a  head  for 
the  sheep;  that  is,  he  would  have  sold  them  for  $5  a  head. 
In  the  second  case  he  would  have  bought  them  for  1  dollar 
more  a  head,  or  for  —  4  dollars;  that  is,  he  would  have 
sold  them  for  4  dollars  a  head. 

When  one  of  the  solutions  is  negative  the  wording  of 
the  problem  may  be  changed,  in  general,  so  as  to  make 
that  solution  positive  and  arithmetically  true. 

Thus,  a  farmer  sold  a  number  of  sheep  for  80  dollars. 
Had  he  sold  4  more  for  the  same  money  he  would  have 
received  1  dollar  a  head  less  for  the  sheep.  How  many 
did  he  sell  ? 

e.g.    1.  The  length  of  a  field  is  12  rods  and  its  breadth  is 


236  QUADRATIC  EQUATIONS. 

10  rods.     How  many  rods  must  be  added  to  the  length  of 
the  field  that  the  area  may  be  100  square  rods  ? 
Let  X  =  number  of  rods  to  be  added. 

Then  (1^^  +  a;)10  =  100. 

10:^  =  100  -  120. 

x=  -2. 

Hence  the  number  of  rods  to  be  added  to  the  length 
is  —  2.  This  is  possible  algebraically,  but  impossible  arith- 
metically. 

In  the  arithmetical  sense,  to  add  means  to  increase; 
and  as  the  area  of  the  field  at  first  was  120  square  rods,  no 
increase  in  its  length  could  make  its  area  100  square  rods. 

But  algebraically,  to  add  —  2  means  to  subtract  2 
arithmetically;  and  were  the  statement,  *' How  many  rods 
must  be  subtracted  from  the  length  of  the  field  to  make  its 
area  100  square  rods  ?"  we  should  find  the  2  to  be  positive 
and,  therefore,  true  in  the  arithmetical  sense. 

e.g.  2.  A's  age  is  40,  and  B's  35.  How  many  years 
hence  will  A's  age  be  twice  B's  ? 

Let  X  =  number  of  years  hence. 

Then  ^0 -\- x  =  2(35  +  x), 

x=  —30. 

This  is  impossible  arithmetically,  but  perfectly  true 
algebraically,  since  —  30  years  hence  means  30  years  ago. 

Had  the  question  been  worded,  "  How  many  years  ago 
would  A's  age  have  been  twice  B's  ?''  the  solution  would  have 
been  positive  and  the  problem  would  have  been  possible 
arithmetically. 

When  imaginary  results  are  obtained  in  the  solution 
of  a  problem,  there  is  either  an  impossibility  in  the  con- 
ditions of  the  problem  or  an  error  in  the  formation  of  the 
equation. 


BOOTS  OF  AN  EQUATION.  237 

e.g.     Divide  12   into  two  parts  whose  product  shall 
be  37. 

Let  X  denote  one  part. 

Then      x{l%  -  x)  =  37. 
nx  -x^  =  37. 
x^  -  12x  +  37  =  0. 
a;2  -  12a;  +  36  -  1  =  0. 

x-Q  ±  V~^^  =  0. 


x 


=  Q-  V  -1,  or  6  +  y  -  1. 


\<^  -  X  -^  Q  -\-  V  -  I,  or  6-4/-1. 

That  is,  12  cannot  be  divided  into  two  parts  whose 
product  is  37. 

EXERCISE  cm. 

I. 

1.  Find  two  numbers  whose  difference  is  7  and 
whose  sum  multiplied  by  the  greater  is  345. 

2.  Find  three  consecutive  numbers  whose  sum  is  equal 
to  3/5  the  product  of  the  last  two. 

3.  Find  two  numbers  whose  difference  is  12  and 
whose  sum  multiplied  by  the  greater  is  560. 

4.  Find  three  consecutive  numbers  whose  sum  is  equal 
to  3/7  the  product  of  the  last  two. 

6.  Find  two  numbers  whose  sum  is  6  and  the  sum  of 
whose  cubes  is  72. 

6.  Find  four  consecutive  numbers  such  that  the  prod- 
uct of  the  last  two  shall  be  equal  to  the  number  composed 
of  the  first  two  used  as  digits. 

7,  Find  four  consecutive  numbers  such  that  the  prod- 


238  QUADRATIC  EQUATIONS. 

uct  of  the  last  two  shall  be  2^  times  the  product  of  the 
first  two. 

II. 

8.  A  merchant  bought  a  quantity  of  flour  for  120 
dollars.  Had  he  bought  10  barrels  more  for  the  same 
money,  the  cost  would  have  been  2  dollars  a  barrel  less. 
How  many  barrels  did  he  buy,  and  at  what  price  ? 

9.  A  merchant  sold  a  quantity  of  wheat  for  16  dollars, 
and  the  loss  per  cent  was  equal  to  the  cost  in  dollars. 
What  was  the  cost  of  the  wheat  ? 

10.  A  merchant  sold  a  quantity  of  cloth  for  96  dollars, 
and  the  gain  per  cent  was  equal  to  the  cost  in  dollars. 
What  was  the  cost  of  the  cloth  ? 

11.  A  crew  can  row  10  miles  down  stream  and  back 
again  in  2  hours  and  40  minutes;  and  the  rate  of  the 
stream  is  2  miles  an  hour.  What  is  the  rate  of  the  crew 
in  still  water  ? 

12.  A  crew  can  row  20  miles  down  stream  and  back 
again  in  7  hours,  and  the  rate  of  the  stream  is  3  miles  an 
hour.     What  is  the  rate  of  the  crew  in  still  water  ? 

173.  Solution  of  the  General  Quadratic    Equation. — 

The  most  general  type  of  a  quadratic  equation  of  one  un- 
known quantity  is 

ax^-\-lx-\-c  =  0.  (A) 

If  we  divide  through  by  a,  then 

x^  4-  -X  A-  ~  =  0', 
a         a 

and  if  we  substitute  j)  for  — ,  and  q  for  — ,  the  equation 

C(/  (t 

becomes 

x'-^px^q  =  0,  (B) 


ROOTS  OF  AN  EQUATION.  239 

which  is  the  quadratic  equation  reduced  to  its  simplest 

form. 

P 
If  in  equation  (B)  we  add  and  subtract  the  square  of  ^, 

we  get 


or  x^ 

which  factors  into 


+^,  +  Z_Z^  =  o, 


Therefore  x  =  l/2(- j9  +  Vp^  -  ^), 


and  l/2(-j9-  Vp^  -  ^). 

On  account  of  the  double  sign  of  the  root  symbol,   y", 
both  values  are  included  in  the  one  expression 


x^l/%{-p±Vf-4.q),  (1) 

which  is  the  solution  of  (B). 

h  c 

If  in  this  equation  we  write  -  for  p  and  —  for  q^  we  have 

a       ^        a 


"  2\     a^^  a"       ar 


1/     ^    ,    Jh^        4«c\ 

or  ^=o ±  r  -T 2"' 

2  \     «  a'^        of  I 

(     h        Vh^  -  4.ac\ 


or  ^  =  o 

.    2\     a 


or  x=  ^i-{—  b  ±  S/IP-  —  ^ac), 

Zci 

which  is  the  solution  of  (A).  (2) 


240  QUADRATIC  EQUATIONS. 

Formulae  (1)  and  (2),  for  the  solution  of  quadratic  equa- 
tions, should  be  so  thoroughly  memorized  that  the  roots  of 
any  quadratic  equation  may  be  written  down  at  sight. 
Formula  (1)  is  most  convenient  for  use  when  the  coefficient 
of  x^  is  unity,  and  formula  (2)  when  the  coefficient  of  x^  is 
not  unity. 

e.g.     1°.  Find  the  roots  of  x^ -{- %x  -  35. 


l/2(-2±  1/4+  140), 
or  l/2(-  2  ±  12). 

Hence  x^  =  5,   and   X2=  —  7. 

2°.  Find  the  roots  of  2x^  -\- 6x  -  12. 


l/4(-  5  ±  4/25  +  96), 
or  l/4(-  5  ±  |/121), 

or  l/4(-  5  ±  11). 

Hence  x^  =  3/2,    and  a^g  =  —  4. 

3°.  Find  the  roots  of  3x^  -{- 7x  -  25. 


l/6(-  7  ±  1/49  +  300), 

or  l/6(-  7  ±  |/349). 

-  7  +  ^^'349  ^  -  7  -  1/349 

Hence  x  = ,     and    Xo  = 


6  '     '  6 

Whether  the  roots  be  rational,  surd,  or  imaginary  de- 
pends upon  the  radicals   Vp^  —  4g  and   Vb'^  —  4ac. 

When  p'^  =  4g  or  P  =  4:ac,  the  roots  are  equal,  since  the 
radical  then  becomes  zero. 

EXERCISE  CIV. 

I. 

1.     x^-\-Qx-\-S  =  0.         2.     x^  -  Ux  -  120  =  0. 
3.     2x^  -  bx^  25.  4.     3x^  -  17a:  +  14  =  0. 


ROOTS  OF  AN  EQUATION.  241 

6.  7^:2  =  22a;  -  15.  6.     {^x  -  'df  =  2a;  +  3. 

7.  a;2_|_|^i8.  8.    x^-^  =  l. 
Sx^       4a;      1       ^ 

10.     (a;  -  2)2  =  1  +  |(a;  +  2). 

11         ^  a;+l       a;  +  2_ 

•  a;  +  l"^a;  +  2'^a;  +  3 

12.  a;2  -|-  2«a;  —  V^  —  a^. 

13.  a;2  +  «(1  -I-  3^')a;  +  3^^^  =  0. 

14.  ax^  +  ^(1  -  a^)x  =  ab\ 


II. 

16.     {a  -  xY  -(a-  x){h  -x)^{x-  If  =  (a  -  b)\ 

16.  a\x —  hy  =  li^{x  —  of. 

17.  (2a- b-  xf  +  9(«  -  ^)2  =  ((«  +  b)-  2x)\ 

,1  ,1  x    ,  a       a       b 

18.  a;  4-  -  =  <?  H .  19.     -  -f  -  =  -  -f  -. 

a;  «  a       X       0       a 

20.     -  +  --—  +  — nr-  =  0. 

a    '  a  -\-  X  ^  a  -\-  2x 


21. 


a;  +  a-]- 2b  _b  —  2a-\-2x 
X  -\-  a  —  2b  ~  b  -\-2a  —  2x 

CO        ^±1    I     ^  +  ^     ,    ^±i   -  Q 

22-     :c  +  2  +  a;4-3"^a;+5  ~  "^^ 

5a;  4-  2      5a;  -  2  _  25a;  -j-  11 
^^-     5.?;  -  2  +  5a;  +  2  ~    5a;  +  2  • 

3a;  +  1       3a;  -  1        9a;  -  13/2 
^*"     3a;  -  1  "^  a;  +  1  ~     3a;  +  1 


242  QUADRATIC  EQUATIONS. 

EXERCISE  CV. 
I. 

1.  Two  trains  run  over  the  same  120  miles  of  rail  with- 
out stopping.  One  of  them  goes  10  miles  an  hour  faster 
than  the  other  and  passes  over  the  distance  in  1  hour  less 
time.     What  is  the  speed  of  the  trains  ? 

2.  Two  trains  run,  without  stopping,  over  the  same  90 
miles  of  rail.  One  of  them  goes  5  miles  an  hour  faster 
than  the  other,  and  passes  over  the  distance  in  15  minutes 
less  time.     What  is  the  speed  of  the  trains  ? 

3.  A  crew  can  row  a  certain  course  up  stream  in  5 
hours,  and  in  still  water  they  could  row  it  in  4^  hours  less 
time  than  it  would  take  them  to  drift  down  stream  to  the 
starting-point.  How  long  would  it  take  them  to  row  back 
with  the  current  ? 

4.  A  crew  can  row  a  certain  course  up  stream  in  6^ 
hours,  and  in  still  water  they  could  have  rowed  it  in  4  hours 
less  time  than  it  would  take  them  to  drift  down  to  the  start- 
ing-point. How  long  would  it  take  them  to  row  back  with 
the  current  ? 

II. 

6.  Simplify  {a'by^y  y\a  -  ^^/'')y\d^H  "  V^)-  \ 

6  Express  a'^h'^^^  -{-  2a^/%  ~ ^'^  without  negative  or 
fractional  exponents. 

7.  Find  the  value  of  (64)"  2/2. 

8.  Divide  «"*  +  ^/^  by  a""  +  *'/'*  and  reduce  the  resulting 
exponents  to  a  single  fraction. 

9.  Multiply  («+^)  by  [0-^^. 
10.     Factor  Ix^  —  Uxy  —  llic  -f  22y. 


ROOTS  OF  AN  EQUATION,  243 

174.  Solution  of  Equations  of  the  Form  of  Trinomial 
duadratics. — Whenever  an  equation  of  one  unknown  quan- 
tity can  be  reduced  to  a  trinomial  the  first  term  of  which 
contains  the  unknown  quantity  only  in  the  square  of  a 
factor,  the  second  term  only  in  the  first  degree  of  the  same 
factor,  and  the  third  term  not  at  all,  it  may  be  first  solved 
as  an  ordinary  quadratic  for  that  factor,  and  then  the  values 
of  the  unknown  quantity  may  be  found  from  values  of  the 
factor. 

e.g.     1°.  Solve     ^{x  -  Sy  -{- 5{x  -  S)  -  2  =  0. 

Factoring,  we  obtain 

{(X  -  d)  -\-2)(d(x  -  '6)  -  1)  =  0; 

and  equating  each  factor  to  zero,  we  have 

a;-  3  +  2  =0,         or    x  =  l; 

and  d{x  —  3)  —  d  —  0,    or    x  =  4. 

2°.  Solve  ex'  -5x^-6=0. 

Factoring,  we  obtain 

(3a;2  +  2)(2a;2-3)  =  0. 

.-.     3a;2  +  2  =  0,    or    x""  =  -2/3; 

and  2x^-3  =  0,    or    x^  =  3/2. 


.'.     x=  ±V  -2/3=  ±1/3V  -i 
and  x=  ±  4/3/2  =  ±  1/2  V6. 

EXERCISE    CVI. 

Solve  the  following  equations  as  quadratics: 
I. 

1.  Q(2x  -  3)2  -  n{2x  -  3)  =  0. 

2,  3a^  -  19^2  4-  20  =  0, 


244  QUADRATIC  EQUATIONS. 

3.  ^x  -  4)2  -  11(^  _  4)  4-  10  =  0. 

4.  (2^2)2  _  7(2:^2)  _|_  12  ==  0. 

6.     {x  -  3)2  _  5(:z;  -  3)  +  6  =  0. 

6.  ^x^  -  33a;2  +  28  ^  0. 

II. 

7.  Ux!^  -  Ux^  +  12  =  0.        8.     64.x^  -  21x^  +  2  =  0. 
9.     8:^6  _|_  37^,3  ^  216.  10.     12x-^-{-x-'  =  d5. 

11.     69-20x-^-x-^  =  0.      12.     a;-4- 21:c-2=:  -  108. 
13.     32ic5  +  l/ic5  =  -  33.         14.     x^  -  3a^/2  =  88. 

EXERCISE   evil. 


1.  A  person  has  12  miles  to  walk.  After  he  has  been 
on  the  road  one  hour  he  increases  his  speed  |  mile  an  hour 
and  finishes  his  journey  in  f  of  an  hour  less  time  than  he 
would  have  accomplished  it  had  he  not  altered  his  speed. 
How  fast  did  he  walk  at  first,  and  how  long  was  he  on  the 
road? 

2.  A  man  has  to  drive  25  miles.  After  he  has  been 
on  the  road  two  hours  he  slackens  the  speed  of  his  horses  1 
mile  an  hour,  and  is  f  of  an  hour  longer  than  he  would 
have  been  had  he  not  changed  the  rate  of  driving.  At 
what  rate  did  he  drive  at  first,  and  how  long  was  he  on  the 
road? 

3J2 

3.  Reduce  3:^2  _  ^_  ^q  ^  single  negative  fraction. 

4.  Rationalize  — — . 

3  -  2  1^5 


WOTS  OF  AN  EQUATION.  245 

II, 

5.  A  and  B  together  can  do  a  piece  of  work  in  a 
certain  time.  Were  each  to  do  half  of  it  alone,  A  would 
have  to  work  2  days  less  and  B  4  days  more  than  when 
they  work  together.    In  what  time  can  they  do  it  together  ? 

6.  A  and  B  can  do  a  piece  of  work  in  a  certain  time. 
Were  each  to  do  half  of  it  alone,  A  would  have  to  work  4 
days  less  and  B  8  days  more  than  when  they  worked 
together.     In  what  time  can  they  do  it  together  ? 


CHAPTER  XX. 

QUADRATIC  EQUATIONS  OP  TWO  UNKNOWN 
QUANTITIES. 

176.  Special  Cases  of  Elimination.  —  Generally,  by 
elimination,  two  equations  of  the  second  degree  with  two 
unknown  quantities  will  produce  an  equation  of  the  fourth 
degree,  which  are  usually  insolvable  by  any  of  the  methods 
yet  given. 

e.g.  x^-}-y  =  a.  (1) 

x-^f=h.  (2) 

From  (1)  we  get  y  =  a  —  x^. 

Substituting  this  in  (2),  we  get 

X  -{-  {a  —  x^Y  =  b, 

or  X  -\-  a^  —  2ax^  -\-  x'^  =  &^, 

which  is  an  equation  of  the  fourth  degree,  and  insolvable 
by  any  of  the  methods  yet  employed. 

There  are,  however,  several  cases  in  which  simultaneous 
quadratics  with  two  unknown  quantities  may  be  solved  by 
the  rules  of  quadratics. 

Case  1°. 

}Vhen  each  of  the  equatioois  is  of  the  form 

ax^  -\-  hy'^  =  c. 

In  this  case  one  of  the  unknown  quantities  may  be 
eliminated  by  addition  or  subtraction,  and  then  the  value 
of  the  other  be  found  by  substitution. 

246 


TWO   UNKNOWN  QXTANTITIES.  247 


e.g.     Solve  the  equations    %x^  +    S?/^  =    56, 

(1) 

4/  -  Ux^  =    12. 

(2) 

Multiplying  (1)  by  4, 

Sx"  +  12/  =r  224. 

Multiplying  (2)  by  3, 

-  39.^2  +  12«/2  =    36. 

Subtracting, 

47a;2  =  igg. 

,*. 

0:2  =  4, 

and 

o:  =  ±  2. 

(3) 

Substituting  (3)  in  (1)^ 

,  we  obtain 
8  +  3^/2  =  56. 

.-. 

3«/2  =  48, 

and 

2/2  =  16. 

.-. 

2/=  ±4. 

Therefore  a:  =  2,  j 

^  =  ±  4;  or  o:  =  —  2,  ?/  = 

±4. 

In  this  case  there 

are  four  possible  sets  of  values  of  x 

and  y  which  satisfy  the  given  equations : 

1.     x  =  2,  y  =  4:.  2.     X  =  2,  y  =  —  4:. 

3.     X  =  —  2,  y  =  4:.  4.     o:  =  —  2,  ^  =  —  4. 

It  would  not  be  correct  to  leave  the  results  in  the  form 
X  =  ±2,  y  =  ±4;  for  this  would  indicate  only  the  first 
and  fourth  of  the  above  sets  of  values. 

EXERCISE  CVIII. 

Solve  the  following  equations: 
I. 

1.     3o:2  +  2tf  =  77,  2.    4o;2  +  Sy^  =  99, 
3/  -  6o:2  =  21.  8o:2  -  12y^  =  23. 

3.     5o:2  _^  4^2  =  170,  4.     o;2  _|_  ^2  ^  io(?>i2  +  n^)^ 
3a:2  _  7^2  ^  _  ge,  o;2-9«/2=  _20?i(3m  -f  4'/^;. 


MS  QUADRATIC  EQUATIONS 

II. 

6.  4:X-16=z  17  Vx.  6.    x^/'  +  a;3/5  =  702. 

7.  Multiply  at  sight  ^-f-  — \-  c  by  j-  -{- c,  and  ex- 
press the  result  without  fractions. 

8.  Factor  5x^  —  lOax  -{-  Sbx  —  16ab. 

Case  2°. 

JVhen  one  equatio7i  is  of  the  second  degree  and  the  other 
of  the  first. 

All  equations  of  this  kind  may  be  solved  by  finding  the 
value  of  one  of  the  unknown  quantities  from  the  first-degree 
equation,  and  then  substituting  that  value  in  the  second- 
degree  equation. 

The  resulting  equation  will  be  a  quadratic  of  one  un- 
known quantity  which  may  be  solved.  When  the  value  of 
one  unknown  quantity  has  been  found  thus,  the  values  of 
the  second  must  be  found  by  substituting  the  values  of  the 
one  already  found  in  the  first-degree  equation. 

e.g.     1.   Solve  the  equations  '6x^  —  xy  =  ^y.         (1) 

2a;  +    y  =  l.  (2) 

From  (2),  we  have  i/  =  7  —  2:r.  (3) 

Substituting  this  value  in  (1),  we  get 

3^2  _  ^(7  _  2x)  =  2(7  -  2a;), 

or  dx^  -7x-{-  2^2  z=  14  -  Ax, 

.'.     5a;2- 3a;- 14  =  0. 

.».     {x  -  2){6x  +  7)  =0. 

Whence     x  =  2,    or    x  =  —  7/5. 

Substituting  these  values  in  (3),  we  get 

y  =  3,    OY   y  =  +49/5. 


OF  TWO   UNKNOWN  QUANTITIES.  249 

Therefore:     1.    ^  =  2,  2/  =  ^• 

2.    x=  -  7/5,     y  =  49/5. 

Certain  examples  in  which  one  equation  is  of  the  third 
degree  and  the  other  of  the  second  degree  may  be  solved  in 
a  similar  way. 

e.g.     2.   Solve  the  equations 

^3  _|_  ^3  =  152^  •  (1) 

X  -\-y  =^.  (2) 

From  (2),  we  obtain        y  =  8  —  x.  (3) 

By  substituting  this  value  of  ^  in  (1),  we  get 
^3  _^  (g  _  ^y  ^  152, 

or  x^  +  512  -  192^;  +  24:X^  -  x^  =  152, 

or  24:X^  -  192x  +  360  =  0, 

or  a;2  _  8a;  4-  15  =  0. 

.-.     {x-5){x-S)  =  0. 
.  •.     a;  =  5,  or  a:  =  3. 
Substituting  these  values  of  x  m  (2),  we  get 

6  +  y=8,  (4) 

and'  3 +  2/ =8.  (5) 

From  (4),  we  have  «/  =  3, 

and  from  (5),  y  =  ^^ 

Therefore     a;  =  5  or   3,  and  ?/  =  3  or  5. 

1.  X  =  5,   y  =  3. 

2.  X  =  3,    y  =  6. 


250  QUADRATIC  EQUATtOm 

EXERCISE  CIX. 

Solve  the  following  equations : 
I. 

1.     3a^  —  xy  =  2y,  2.     x -\-  y  =  —  2, 

2x  -{-  y  =  7,  xy  =  —  24:. 

5.  X  —  y  =  2,  ^.     x^  -{-  xy  —  y^  =  —  11, 
x^  -\-  y^  =  34:,  ^  —  y  =  —  4:. 

6.  x^  -y^=  -  296,  6.     x^ -{- y^  =  152, 
X  —  y  =  —  2.  X  -\-  y  =  8. 


II. 

7.     x-y  =  l, 
xy  =  a^  -\-  a. 

2A  +  3/y  =  1. 

9.        8^  -y^=    -7, 

2x-y=-l. 

10.     x/y-^y/x  =  l0/3, 
3x-2y=-  12. 

11.     2a;2/«  +  3:cV»  -  56 

=  0. 

12.     Factor  Ibax  —  lOx  +  6«J  —  4 J. 

Case  3°. 

An  expression  is  said  to  be  symmetrical  with  respect  to 
any  of  its  letters  when  any  two  of  them  can  be  interchanged 
without  altering  the  value  of  the  expression. 

e.g.  The  expression  ab  -\-  be  -{-  ca  is  symmetrical  with 
respect  to  the  letters  a,  b,  and  c;  for  if  any  two  of 
them,  as  a  and  b,  be  interchanged,  the  expression  becomes 
ba  -\-  ac  -\-  cb,  which  is  the  same  as  the  original  expression 
in  meaning. 

The  equations        x  -\-  y  =  2, 

xy  =  3, 
are  symmetrical  in  x  and  y. 


OF  TWO   UNKNOWN  QUANTITIES.  251 

The  equations        x  —  y  =  a, 

xy=  I, 

are  symmetrical  except  in  their  signs. 

When  tlie  given  equations  are  sym7netrical  in  x  and  y, 
and  one  of  them  is  of  the  second  degree  and  the  other  of  the 
first.)  they  may  he  solved  hy  combining  them  in  such  a  luay 
as  to  oMain  the  values  of  x  -\-  y  and  x  ~  y. 

e.g.  Solve  the  equations  x-\-  y  =  1,  (1) 

xy=-Q.  (2) 

Squaring  (1),  we  have      x^  +  '^^y  +  ^^  =  !•  (3) 

Subtracting  4  times  (2)  from  (3),  we  get 

x^  —  2xy  -\-y^  =  25, 

which  is  the  square  oix  —  y. 

Extracting  the  square  root  of  each  member, 

X  —  y  =  ±  5.  (4) 

Adding  (4)  to  (1),  we  have 

2x  =  Q   or    —  4. 
a;  -  3   or    -  2. 
Subtracting  (4)  from  (1),  we  have 

2y=  -4:  or   6. 
y=  -2   or   3. 
1.     x  =  3,     y=-2. 
.',     2.     x=~2,     y  =  d. 

This  method  may  be  used  in  many  cases  when  the  equa- 
tions are  symmetrical  except  with  respect  to  the  signs  of  the 
terms. 

e.g.  Solve  the  equations  x^  -\-  y^  =  G5,  (1) 

X  -  y  =  -3.  (2) 


252  QUADRATIC  BQUATIOm 

Multiply  (1)  by  2,  and  subtract  the  square  of  (2)  from 
the  result: 

2x^  +  2?/^  =130 
x^  —  %xy  -f  1/2  =  9 
a;2 •+  ^xy  -^  y'^  =  121 
.-.       x^y=±ll.  (3) 

Add  (3)  to  (2),  and  we  get 

2^  =  8     or     -  14. 
.  •.     X  =  4:    or     —    7. 

Subtract  (3)  from  (2),  and  we  get 

—  2?/  =  —  14    or    8. 
«/  ==  7     or     —  4. 
1.     X  =  4:,     y  =  7. 
.-.     2.     x^-1l,     «/=-4. 

Certain  examples  in  which  one  equation  is  of  the  third 
degree  and  the  other  is  of  the  first  or  second  may  be  solved 
by  the  methods  of  this  case. 

e.g.  Solve  the  equations  x^  -\-  y^  =  189,  (1) 

x^-xy-\-y^^  21.  (2) 

Divide  (1)  by  (2),  and  we  get 

^  +  2/ =  9.  (3) 

Square  (3)  and  subtract  (2)  from  the  result: 

x^  H-  ^xy  +  y^  =  81 

x^  —    xy  -\-  y^  =  21 

3xy  =  60 

.-.     -  xy=  -  20.  (4) 


OF  TWO   UNKNOWN  QUANTITIES.  253 

Add  (4)  to  (2),  and  we  get 

x^  —  %xy  -[-  ?/^  =  1. 
,-,     x-y=±\.  (5) 

Add  (5)  to  (3),  and  we  get 

1x  =  10    or  8. 

.'.     x=    5     or  4. 

Subtract  (5)  from  (3),  and  we  get 

2y  =  S     or  10. 

.'.     y  =  4:    or  5. 

1.     X  =  5,  2/  —  ^• 

.-.     2.     X  =  4:,  y  =  5. 

In  solving  examples  under  this  case,  it  must  be  borne  in 
mind  that,  in  every  instance,  we  must  combine  the  given 
equations  in  such  a  way  as  to  obtain  the  values  of  x  -\-  y 
and  X  —  y. 

e.g.  Solve  the  equations  x^  -{-  y^  =  13,  (1) 

xy=    6.  (2) 

Multiply  (2)  by  2,  and  add  the  result  to  (1),  and  also 
subtract  it  from  (1),  and  we  get 

x^  +  2xy  +  y^  =  25, 

and  x^  —  2xy  +  /  =    1. 

.'.     x  +  y  =  ±  5, 

and  X  —  y  =  ±  1. 

.-.  2a;  =  5  ±  1     or     -  5  ±  1. 

.*.     X  =  3     or     2,     or     —  2,     or     —  3. 

And  2^  =  5  =F  1     or     -5^1. 

.  *.     ?/  =  2     or     3,     or     —  3,     or     —  2. 


254  QUADRATIC  EQUATIONS 

Therefore;         1.  x  =  d,  ?/  =  3. 

2.  2-  =  2,  y  =  '6. 

3.  x=  -%  y  =  -3. 

4.  X  =  -  3,  y  =  -  2. 

A  few  examples  in  which  both  equations  arc  of  the  third 
degree  may  be  solved  by  the  methods  of  this  case. 

e.g.     1.  Solve  the  equations  x^  —  y^  —  26,  (1) 

':i?y  —  xy^  —  6.  (2) 

Multiply  (2)  by  3  and  subtract  the  result  from  (1),  and 
we  get 

x^  -  3x^y  +  3a;?/2  -^3^8.  (3) 

Extract  the  cube  root  of  (3),  and  we  get 

x-y  =  ^.  (4) 

Divide  (2)  by  (4),  and  we  get 

xy  =  3.  (6) 

From  (4)  and  (5),  we  get  a;  -f- 1/  =  ±4.  (6) 

.-.     2a;  =6     or     -2, 


and 

X  ■- 

=  3     or     - 

1. 

Also, 

"^y- 

=  2     or     - 

6. 

.'.     y: 

=  1     or     - 

3. 

Therefore : 

1. 

x  =  3, 

y  =  l. 

2. 

x=-l, 
EXERCISE 

y  =  -d. 
;  ex. 

Solve  the  following  equations; 

1.     xy  =  42, 
x-{-y  = 

13. 

1. 
2 

xy  ~  24, 

x  +  y  "• 

11. 

OF  TWO   UNKNOWN  QUANTITIES.  255 

8.  x^  +  if  =  29,  4.     x^  +  ?/^  =  58, 

X  -\-  y  =  1.  X  -{-  y  =  10. 

6.  x^-\-y'^=  26,  6.     x^-\-y'^  =  68, 
X  —  y  =  —  A:.  X  —  y  =  —  Q. 

7.  xy  =  —  18,  8.     xy  =  —  72, 

a;  —  ?/  =  11.  a;  —  ?/  =  —  18. 

9.  x^-y^  =  279,  10.     a;3  _|_  ^3  ^  152^ 

x^  -{-  xy  -{-  y^  =  93.  x^  —  xy  -\-  y^  =  19. 

11.     x^-y^=  152,  12.  a;3  -f  ?/3  ^  637, 

X  —  y  =  2.  X  -\-  y  =  Id. 
II. 

13.     a;3  +  ?/3  =  243,  14.  rc3  -y^^  386, 

a;^?/  +  :ci/^  =  162.  x^y  —  xy^  =  126. 

15.  x^-y^  =  Wb  +  W, 
xy(x  —  y)  =  2b(d'^  —  b^). 

16.  x'-^xy-\-y^  =  Id'  -  Idab  +  W, 
x?-xy^y^  =  Za^-  dab  +  Sb\ 

Case  4°. 

An  expression  is  said  to  be  homogeneous  when  each  of 
its  terms  is  of  the  same  degree. 

Certain  equations  which  are  of  the  form :  a  homogeneoits 
expression  in  x  and  y  of  the  second  degree  equals  a  constant, 
may  be  solved  by  the  methods  of  cases  1°  a7id  3°.  When 
such  equations  can  be  solved  by  neither  of  these  methods, 
they  may  be  solved  by  putting  y  =  mx,  and  solving,  first 
for  m,  then  for  x,  and  finally  for  y. 

e.g.   Solve  the  equations  a;^  —  %xy  =  —  8.  (1) 

x'-\-   y'=rd.  (2) 

Putting  y  =  mx,  we  have 

x^  -  2mx^  =  -S,    or    x'  =  —^—-,  (3) 

27n  —  1  ^  ' 


256  QUADRATIC  EQUATIONS 

1  ^ 
and  x^  +  wV  =  13,      or    x"  =  ,         , 

1  +  m^ 

8  13 


2m  —  1      1  -{-  m^ 
.-.     8  +  8m2  =  26m  -  13, 
or  8m2  -  26m  +  21  =  0, 

or  (2m  -  3) (4m  -  7)  =  0. 

.-.     m  =  3/2     or     7/4. 
Substituting  the  first  of  these  values  in  (3),  we  get 

x^  -      ^      -  4 

.-.     X  =  ±  2. 
Substitute  these  values  of  x  in  (2),  and  we  get 

^  =±3. 
Substituting  the  second  value  of  m  in  (3),  we  get 

a_  8      _16 

^  ~  7/2  -  1  ~  5  * 

.'.  '  X  =  ±  4/5  1/5. 
Substitute  these  values  in  (2),  and  we  get 

y  =  ±1/b  V6. 
Then:  1.     x  =  4^/5  V5,     y  =  l/bVb, 

2.  X  =.^/bVb,     y=-  7/5  V6. 

3.  ic  =  -  4/5  4/5",     y  =  7/5  \''5. 

4.  X  =  -  4/5  V5,     y  =  -  7/5  Vb. 

In  each  case  the  value  of  y  might  have  been  obtained 
by  substituting  the  values  of  m  and  xm  y  =  mx. 


OF  TWO   UNKNOWN  QUANTITIES.  257 

EXERCISE  CXI. 

Solve  the  following  equations : 
I. 
1.    a:2  _^  3^^  ^  28,  2.     x^-\-xy-\-  2tf  =  74, 

xy  +  %2  =  8.  2a;2  +  2xy  +  y^  =  73. 

3.  x^  -\-  xy  —  6y^  =  24, 
2^2  -j-  3xy  -  lOy^  =  32. 

II. 

4.  a;2  +  a;^  —  6?/2  =  21,  5.     ^^  —  »;«/  +  2/^  =  ^1* 
a:?/  —  2?/^           =4.  2/^  —  2a;?/  =  —  15. 

6.    x^  +  xy  +  2?/''^  =  44, 
2ay^-xy-\-y^=  16. 

EXERCISE  CXII. 
I. 

Solve  the  following  problems  by  using  two  unknown 
quantities : 

1.  The  sum  of  two  numbers  is  8,  and  the  sum  of  their 
squares  is  34.     What  are  the  numbers  ? 

2.  The  difference  of  two  numbers  is  3,  and  the  differ- 
ence of  their  squares  is  33.     What  are  the  numbers  ? 

3.  The  sum  of  the  squares  of  two  numbers  is  106,  and 
the  product  of  the  numbers  is  45.    What  are  the  numbers  ? 

4.  The  difference  of  two  numbers  is  6,  and  their  prod- 
uct is  40.     What  are  the  numbers  ? 

6.     The  sum  of  two  numbers  is  7,  and  the  sum  of  their 
cubes  is  91.     What  are  the  numbers  ? 

6.  The  difference  of  two  numbers  is  4,  and  the  differ- 
ence of  their  cubes  is  316.    What  are  the  numbers  ? 

7.  rind  two  numbers  such  that  the  square  of  the  first 


258  QUADRATIC  EQUATIONS. 

and  twice  the  square  of  the  second  shall  together  equal  32, 
and  the  square  of  the  second  and  three  times  the  product 
of  the  two  shall  equal  27. 

II. 

8.  Find  two  numbers  such  that  three  times  the  square 
of  the  smaller  and  the  square  of  the  larger  shall  together 
equal  7,  and  the  square  of  the  smaller  shall  be  7  less  than 
four  times  the  product  of  the  two. 

9.  A  man  bought  8  cows  and  5  sheep  for  255  dollars. 
He  bought  3  more  sheep  for  39  dollars  than  cows  for  300 
dollars.     What  was  the  price  of  each  ? 

10.  A  number  is  composed  of  two  digits.  If  its  digits 
be  inverted,  the  sum  of  the  new  and  original  numbers  will 
be  44,  and  their  product  403.     What  are  the  numbers  ? 

11.  Multiply  a  A -,  by  J r—i- 

^•^  a  —  0    ^  a  -\-  b 

12.  Factor  l^x^  —  Sxy  —  9x^y^  +  6y^. 

13.  Reduce  —  ^r^-^  -|-  75—  to  a  single  negative  fraction. 

obd         oCi 

14.  Simplify  {l/VZb)-y\ 

15.  Multiply   Vl  by  V^. 

16.  Express  the  following  without  fractional  or  nega- 
tive indices : 

^2/3^-1  _  a-y^b. 

8-51^ 


'•17,     Rationalize  the  denominator  of 


3-21^2 


CHAPTER  XXL 

INDETERMINATE  EQUATIONS  OP  THE  FIRST 
DEGBEE. 

176.  Indeterminate  Equations. — Equations  are  inde- 
terminate when  the  number  of  independent  equations 
given  is  less  than  that  of  the  unknown  quantities  which 
they  contain.  For  when  such  equations  are  solved  for  any- 
one of  their  letters,  the  value  obtained  will  contain  con- 
stants and  one  or  more  of  the  letters  which  represent  the 
other  unknown  quantities.  Hence  the  value  of  the  letter 
found  will  vary  with  the  value  assigned  to  the  other 
letters. 

Thus,  if  2a;  -|-  5«^  =  8,  a:  =  4  —  5/2?/,  and  y  may  take 
as  many  values  as  we  please,  and  to  every  value  of  y  will 
correspond  a  single  value  of  x;  and,  conversely,  to  every 
value  of  X  will  correspond  a  single  value  of  y.  Unless 
some  restrictions  be  placed  on  the  values  of  the  unknown 
quantities,  the  equation  may  be  satisfied  in  an  indefinite 
number  of  ways. 

If,  however,  the  values  of  the  unknown  quantities  are 
subject  to  any  restriction,  n  equations  may  suffice  to 
determine  the  values  of  more  than  n  unknown  quantities. 

In  the  present  chapter  we  shall  consider  only  indeter- 
minate equations  of  the  first  degree  in  which  the  values  of 
the  unknown  quantities  are  restricted  to  positive  integers. 

177.  Solution  of  Indeterminate  Equations  of  the  First 
Degree  in  x  and  y. — Every  equation  of  the  first  degree  in 
X  and  y  may  be  reduced  to  the  form  ax  ±  by  =  ±  c,  in 

859 


260  INDETERMINATE  EQUATIONS 

which  ay  ^,  and  c  are  positive  integers,  and  have  no  com- 
mon factor. 

The  form  ax  -\-  by  =  —  c  cannot  be  solved  for  positive 
integers ;  for  it  a,  b,  x,  and  y  are  positive  integers,  ax  -f-  by 
must  also  be  a  positive  integer. 

The  remaining  forms,  ax  ±by  =  c  and  ax  —  by—  —  c, 
cannot  be  solved  for  positive  integers  when  a  and  b  are 
commensurable.  For  if  x  and  y  are  positive  integers,  the 
common  factor  of  a  and  b  must  also  be  a  factor  of  ax  +  by, 
and  therefore  of  c,  which  contradicts  the  hypothesis  that 
a,  b,  and  c  have  no  common  factor. 

The  form  ax  —  by=  —  c  becomes  by  changing  its  signs 
by  —  ax  =  c,  which  is  essentially  the  same  as  ax  —  by  =  c, 
a  and  b  and  x  and  y  being  interchanged. 

Hence  the  two  type  forms  ax-\-by  =  c  and  ax  —  by  —  c 
are  the  only  ones  that  need  be  considered,  and  those  only 
in  the  cases  in  which  a  and  b  are  prime  to  each  other. 

Ex.     Solve  hx  -f-  12y  =  263  in  positive  integers. 

Divide  through  by  5,  the  smaller  coefficient,  and  we  get 

^  +  ^y  +  ^=52  +  |. 

.-.     :.  +  2^  +  ?^^  =  52.  (1) 

Since  x  and  y  are  both  integers,  and  the  whole  of  the 
first  member  is  an  integer,  therefore 

-=^-r —  =  an  integer. 

Multiplying  this  fraction  by  the  integer  which  will  make 
the  coefficient  of  y  one  more  than  the  denominator  (5),  or 
than  a  multiple  of  the  denominator,  we  get 

6?/  -  9 

-^^—^ —  =  an  integer; 


OF  THE  FIRST  DEGREE.  261 

11  —  4 
that  is,  y  —  1  -\-  '^—z —  =  an  integer. 

V  —  4  .   , 

.  •.     —— —  =  an  integer  =  p. 
o 

.'.     y  —  4:  =  5p, 

or  y  =  5p  -\-  4:.  (2) 

Substituting  this  vahie  of  y  in  (1),  we  get 

0 

or  x+  lOp  -\-S  +  2p+l  =  52, 

or  x-\-12p  =  43. 

.-.     X  =  4:3 -Up.  (3) 

From  (2)  and  (3)  it  is  evident  that  x  a^nd  y  will  be 
integral  when  p  is  an  integer  and  only  when  p  is  an 
integer;  for  they  will  both  be  integers  when  5p  and  12p 
are  both  integers  and  in  no  other  case,  and  6p  and  12^0  will 
be  integral  when  p  is  integral  and  in  no  other  case. 

From  (3)  it  is  evident  that  x  will  be  negative  when  p 
exceeds  3,  and  y  will  be  negative  when  p  is  negative. 
Hence  p  must  be  a  positive  integer  less  than  4.  Hence 
the  only  possible  values  of  p  are  0,  1,  2,  3.  Thus  the 
only  positive  integral  values  of  x  and  y  are  obtained  by 
putting  in  (2)  and  (3)  p  =  0,  1,  2,  and  3. 

The  corresponding  values  of  x  and  y  are  shown  in  the 
following  table: 

p  =  0,l,  2,  3, 

X  =  43,  31,  19,  7, 

.      2/  =  4,  9,  14,  19. 

Note  that  the  coefficients  ofp  in  the  values  of  x  and  y 
in   (2)   and   (3)  are  the  coefficients  of  y  and  x  respect- 


262  INDETERMINATE  EQUATIONS 

ively  in  the  given  equation,  and  that  one  of  the  signs  is 
changed. 

Hence  when  the  given  equation  has  the  type  form 
ax  -\-  by  =  c,  the  term  in  p  in  the  value  of  x  ov  y  must  be 
negative,  and  the  integral  values  of  p  and  therefore  of 
X  and  y  must  be  limited. 

Ex.    2.  Solve  Sx  —  Sy  —  28  in  positive  units. 
Dividing  by  3,  the  smaller  coefficient,  we  get 

,.+  !_, .9  +  1. 
.-.     2x-y  +  ^^^  =  9.  (1) 


2x-l 

— - —  =  an  integer. 


Multiplying  by  2  so  as  to  make  the  coefficient  of  x 
greater  by  one  than  3, 

4:X-2 

—  an  integer. 


3 


■x-2 
2 


X  ~\ —  =  an  integer. 

o 


an  integer  =  p. 


3 

.-.     X  —  "2  =  3p, 
or  x  =  ?>p  -h  2.  (2) 

Substituting  this  value  of  x  in  (1),  we  get 

or  4  +  6iJ-y+l  +  2jo  =  9, 


OF  THE  FIRST  DEGREE.  263 

or  8p  —  y  =  4:. 

.:      y  =  8p-i.  (3) 

From  (2)  and  (3)  we  see  that  p  may  be  any  positive 
integer  except  zero. 

When  p  =  1,  2,  S,  etc., 

:zj  =  5,  8,  11,  etc., 

and  y  —  ^>  1^?  20,  etc. 

In  this  case  the  term  in  p  is  positive  in  both  (2)  and 
(3),  and  the  number  of  solutions  is  unlimited.  This  will 
be  the  case  always  when  the  equation  has  the  type  form 
ax  —  hy  =  c. 

178.  Solution  of  Indeterminate  Equations  of  the  First 
Degree  in  x,  y,  and  z. — To  solve  two  equations  in  three 
unknown  quantities  for  positive  integers:  first  eliminate 
one  of  the  unknown  quantities  so  as  to  get  one  equation  in 
two  unknown  quantities;  then  solve  this  for  positive 
integers  and  obtain  the  value  of  each  of  the  two  unknown 
quantities  in  terms  of  p  and  constants;  and  finally  sub- 
stitute these  two  values  in  one  of  the  original  equations  to 
find  the  value  of  the  third  unknown  quantity  in  terms  of 
m  and  a  constant,  observe  what  values  of  p  will  make  each 
of  these  three  positive  integers,  and  find  the  corresponding 
values  of  each  of  the  unknown  quantities. 

e.g.     Solve      2a;  +  3?/  -  5^  =  -  8, 

bx-   y-\-4:Z  =  21,  (1) 

for  positive  integers. 

Eliminating  y  by  addition,  we  get 

17:^:4-7^  =  55.  (2) 

...     2^  +  .  +  ^=7  +  |, 


264 


INDETERMINATE  EQUATIONS 


or 


o      i        ,    3:?^  -  6       ^ 


3a:  ~  6       ,   ^ 
— —  =  integer. 


15:?;  -  30 


2^-4  + 


=  integer, 


—  integer. 


x-2 


integer  =  p. 


x-2  =  7p, 

or  x  =  7p-^2.  (3) 

Substituting  this  value  of  x  in  (2),  we  get 
119jo  +  34  +  7;?  =  55, 
or  119;?  +  7;2  =  21. 

.%     17i?.+  ^  =  3. 

.-.     ;2  =  3-17i?.  (4) 

Substituting  (3)  and  (4)  in  (1),  we  get 

35J9+ 10  -  2/ +  12  -  68j9  =  21, 
or  —  'SSp  —  y  =  —  1, 

y  =  l-  33p.  (5) 

The  only  value  of  p  that  can  make  z  a  positive  integer 
is  0.     Substitute  this  value  in  (3),  (4),  and  (5),  and  we  get 

x  =  2, 
and  z  =  d. 


OF  THE  FIRST  DEOBEE.  265 

EXERCISE  CXIII. 

Solve  the  following  equations  in  positive  integers: 

I. 

I.  7a; +15?/=    59.  2.       8a;  +  13.y  =  138. 
3      7^;  -f    9^  =  100.  4.     13a;  +  lly  =  200. 

5      Find  the  number  of  solutions  in  positive  integers  of 

11a;  +  15^  =  1031. 

Solve  the  following  equations  in  positive  integers : 

6.       Qx -\- 7y -\- 4:z  =  122,        7.     12x  -  Uy -\- 4.z  =  22, 
11a;  -{-Sy  -6z  =  145.  -  4a;  +    6y -\-    z=17. 

II. 

8.     20a;  -  21y  =  38,  9.      7a;  +  4^  +  19z  =  84. 

3^  +    4^  =  34. 

10.     23a;  +  17y  +  11^  =  130. 

Find  the  general  integral  solutions  of  the  following 
equations : 

II.  7a;  -  Idy  =  15.  12.     9a;  -  lly  =  4=. 
Solve  in  least  positive  integers : 

13.     119a;  -  105y  =  217.  14.     49a;  -  69y  =  100. 

15.  How  can  a  length  of  4  feet  be  measured  by  means 
of  two  measures,  one  7  inches  long  and  the  other  13  inches 
long? 

16.  How  can  45  pounds  be  exactly  measured  by  means 
of  4-pound  and  7-pound  weights  ? 

17.  In  how  many  different  ways  can  the  sum  of  $3.90 
be  paid  with  fifty-  and  twenty-cent  pieces  ? 


266  INDETERMINATE  EQUATIONS. 

18.  In  how  many  different  ways  can  the  sum  of  $5.10 
be  paid  with  half-dollars,  quarter-dollars,  and  dimes,  the 
whole  payment  to  be  made  with  twenty  pieces  ? 

19.  A  farmer  purchased  a  number  of  pigs,  sheep,  and 
calves  for  160  dollars.  The  pigs  cost  3  dollars  each,  the 
sheep  4  dollars  each,  and  the  calves  7  dollars  each ;  and  the 
number  of  calves  was  equal  to  the  number  of  pigs  and 
sheep  together.     How  many  of  each  did  he  buy  ? 

20.  Find  the  least  multiples  of  23  and  15  which  differ 
by  2. 

21.  Find   two  fractions  whose  denominators  shall  be 

113 
respectively  9  and  5  and  whose  sum  shall  be    .  ^  . 


CHAPTER  XXII. 
INEQUALITIES. 

179.  Definition  of  Greater  and  Less  Quantities. — One 

quantity  is  said  to  be  greater  than  another  when  the  remain- 
der obtained  by  subtracting  the  second  from  the  first  is 
positive;  and  one  quantity  is  said  to  be  less  than  another 
when  the  remainder  obtained  by  subtracting  the  second 
from  the  first  is  negative. 

N.B. — Throughout  the  present  chapter  every  letter  is 
supposed  to  denote  a  real  positive  quantity,  unless  the  con- 
trary is  stated. 

In  accordance  with  the  definition  just  given  a  is  greater 
than  J)  when  a  —  h  \^  positive,  and,  conversely,  when  a  is 
greater  than  h,  a  —  h  m  positive.  Also,  a  is  less  than  h  when 
«  —  J  is  negative,  and,  conversely,  when  a  is  less  than  h, 
a  —  J  is  negative.  Thus  2  is  greater  than  —  3  because 
2  —  (—  3),  or  5,  is  positive;  also  —  2  is  greater  than  —  3 
because  —  2  —  (—  3),  orl,  is  positive.  Again,  —  2  is 
less  than  1  because  —  2  —  1,  or  —  3,  is  negative;  and  —  4 
is  less  than  —  2  because  —  4  —  (—  2),  or  —  2,  is  nega- 
tive. 

According  to  this  definition  zero  must  also  be  regarded 
as  greater  than  any  negative  quantity. 

180.  Inequalities. — An  inequality  is  an  algebraic  state- 
ment of  the  fact  that  one  of  two  expressions  is  greater  than 
the  other.  The  two  expressions  compared  are  connected 
together  by  the  sign  >,  "greater  than,  ^' or  <,  "less  than," 

267 


268  INEQUALITIES. 

the  open  end  of  the  symbol  always  being  directed  towards 
the  larger  member  of  the  inequality. 

Two  or  more  inequalities  are  said  to  be  in  the  same 
sense,  or  of  the  same  species,  when  the  first  member  of  each 
is  the  greater  or  the  less,  and  two  inequalities  are  said  to  be 
in  the  opposite  sense,  or  of  the  opposite  species,  when  the 
first  member  of  the  one  is  the  greater,  and  of  the  other  is 
the  less. 

Thus  a  >  h  and  c  >  d  are  two  inequalities  in  the  same 
sense,  or  of  the  same  species.  So  also  are  m<n  and  2^<q. 
But  a  >  h  and  c  <  d,  or  m  <  n  and p>  q  are  inequalities 
in  the  opposite  sense,  or  of  opposite  species. 

The  working  rules  for  inequalities  differ  in  some  re- 
spects from  those  for  equations.  They  are  based  upon  cer- 
tain elementary  theorems  of  inequality  which  are  readily 
deduced  from  the  axioms  of  equality. 

Theoeem  I.  If  equals  be  added  to  unequals,  the  sum 
will  be  unequal  in  the  same  sense. 

Let  a  >  b,  and  let  their  difference  be  denoted  by  r. 
Then 

a  =  b  -\-  r. 

Adding  x  to  each  member  of  this  equation,  we  get 

a-\-x  =  b-\-x-\-r. 

.'.     a  -\-  X  >  b  -{-  X. 

Theorem  II.  If  equals  be  tahenfrom  unequals,  the  re- 
7nainders  will  be  unequal  i7i  the  same  se^ise. 

Let  a  >  b,  and  let  their  difference  be  denoted  by  r. 
Then 

a  =  b  -{-  r. 

Subtracting  x  from  each  member  of  this  equation,  we 
get 

a  —  X  =  (b  —  x)  -\-  r. 

,\     a  —  x  >  b  —  x. 


INEQUALITIES.  269 

Cor.  From  these  two  theorems  it  follows  that  we  have 
the  right  to  add  equals  to  the  members  of  an  inequality, 
and  to  subtract  equals  from  the  members  of  an  inequality, 
without  altering  the  sign  of  inequality. 

Also,  that  we  have  the  right  to  transfer  a  term  from  one 
member  of  an  inequality  to  the  other  by  changing  its  signs, 
without  altering  the  sign  of  inequality. 

Theorem  III.  If  imequals  be  suUracted  from  equals, 
the  remainders  loill  he  unequal  in  the  reverse  sense. 

Let  a>  by  and  let  their  difference  be  denoted  by  r. 
Then 

a  =  b  -\-  r. 

Subtracting  each  member  of  this  equation  from  x,  we 
get 

X  —  a  =  X  —  {b  -{-  r)  =  {x  —  b)  —  r, 
.'.     X  —  a  <x  ~b. 

Cor.  If  x  =  0,  we  would  have  —  a  <  —b.  Hence 
when  we  reverse  the  signs  of  an  inequality,  we  must  also 
reverse  the  sign  of  inequality. 

Theorem  IV.  If  unequals  be  multiplied  by  equals,  the 
products  luill  be  unequal  in  the  same  sense. 

Let  ay  b,  and  let  their  difference  be  denoted  by  r. 
Then 

a  =  b  -\-r. 

Multiplying  both  members  of  this  equation  by  x,  we  get 
ax  =  bx-\-  rx. 
.  *.     ax  >  bx. 

Theorem  V.  If  unequals  be  divided  by  equals,  the 
quotients  will  be  unequal  in  the  same  sense. 

Put  a  =  b  -\-  r  2^%  heretofore. 


270  INEQUALITIES. 

Dividing  each  member  of  this  equation  by  x,  we  get 
a  _h       r 

X  ~  X        x' 

a      h 

Cor.  From  Theorems  IV  and  V  it  follows  that  we 
have  the  right  to  multiply  or  divide  both  members  of  an 
inequality  by  the  same  positive  quantity  without  altering 
the  sign  of  inequality. 

If,  however,  both  members  of  an  inequality  be  multi- 
plied or  divided  by  a  negative  quantity,  the  signs  of  both 
members  will  be  reversed.  This  reversal  of  signs  is  equiv- 
alent to  an  interchange  of  the  members,  and  therefore  it 
reverses  the  character  of  the  inequality.  Hence,  on  such 
multiplication,  the  sign  of  inequality  must  be  reversed. 

Theorem  VI.  If  equals  ie  divided  iy  unequals,  the 
quotients  will  he  unequal  in  the  opposite  sense. 

Put  as  before  a  =  h  -\-r. 

Dividing  x  by  each  member  of  this  equation,  we  get 


X 

X               OX           ox  -j-  rx  —  rx 

a  ~ 

~  b-i-r~b{b-]-  7')~     b{b  +  r) 

x{b  -{-  r)            rx 
-b{b  +  r)-  b{b^r) 

X           rx 

~b      b{b-^  r)' 

X         X 

Theorem  VII.  If  two  inequalities  of  the  same  species 
be  added  together,  the  results  will  be  unequal  in  the  same 
sense. 

Let  a>  b  and  c  >  d. 


INEQUALITIES.  271 

Put  a  =  ^  -|-  r,  and  c  =  d  -{•  s. 
Then,  by  addition  of  equals, 

a-\-c  =  b-\-d-\-r-{-s, 

.'.     a  -\-  c  >  b  -\-  d. 

Note. — By  subtraction  we  would  get 

a  —  c  =  b  —  d-\-r  —  s\ 

from  which  we  cannot  infer  whether  «  —  c  >  6  —  t?,  or 
a  —  c  <b  —  d. 

If  r  >  s,  a  —  c  >  b  —  d;  but  it  r  <  s,  a  —  c  <  b  —  d. 

Hence  addition  of  corresponding  members  of  inequali- 
ties of  the  same  species  without  changing  the  sign  of  in- 
equality is  always  admissible,  but  not  subtraction. 

CoE.     It  a  >  b,  0  >  dj  e  >  f,  etc.,  then 

a  -{-  c  -\-  e  -{■  etc.  >  b  -\-  d  -\- f -\-  etc. 

Theorem  VIII.  If  two  mequalities  of  the  same  species 
be  multiplied  together,  the  results  will  be  unequal  iu  the 
same  sense. 

Let  a>  b,  and  c  >  d. 

Put  a  =  b  -\-  r,  and  c  =  d  -{■  s. 

Then,  by  the  multiplication  of  equals, 

ac  =  (b  -\-  r)(d  -j-  s)  =  bd  -\-  bs  -\-  dr  -{■  rs, 

.'.     ac  >  bd. 

Cor.  1.     It  a  >  b,  c  >  d,  e  >  f  etc.,  then 

a  .  c .  e  .  etc.  >  b  .  d  .f.  etc. 

Cor.  2.     It  a  >  b,  then  a'"  >  b"^. 
Cor.  3.     If  «  >  b,  then  a-"'  <  b-"^, 

EXERCISE  CXIV. 

1.    For  what  values  of  x  is 

5^--<  — +  6? 


272  INEQ  UALITIE8. 

Multiplying  both  members  by  5,  we  get 

^bx  -  16  <  10:?;  +  30. 

By  transposition,     15a;  <  46. 

.-.     x<^^. 

This  inequality  holds  for  all  values  of  x  less  than  S^^g 

2.    For  what  values  of  x  and  y  are 

4:c  +  3?/  >  27, 

3a;  +  4^  =  29  ? 

Multiplying  both  members  of  the  inequality  by  4,  and 
of  the  equation  by  3,  we  get 

16a;  +  12?/ >  108; 

9a; +  12?/=    87; 

.-.      7a;  >    21; 

.-.       X  >      3. 

Multiplying  both  members  of  the  inequality  by  3,  and 
of  the  equation  by  4,  we  get 

12a;  +  9?/>  81; 
12a;  +  16?/  =  116; 
"    7?/  >  -  35. 

.-.      7?/<35. 

.-.        ?/<5. 

Hence  the  values  are  all  of  those  of  x  greater  than  3, 
and  of  y  less  than  5,  which  make  3a;  +  4?/  ==  29. 

N.B. — The  values  of  x  and  y  obtained  as  above  are 
called  the  limits  of  x  and  y.  That  is,  they  are  the  values 
which  bound  the  possible  values  which  x  and  y  can  have 
under  the  given  conditions. 


INEQUALITIES.  273 

Find  the  limits  of  x  in  the  following  cases : 

3.  (42;  +  2)2-29>  (22;  +  2)(8a:-4). 

4.  Cdx  -  2)(4^  +  3)  >  {'^x  -  ^){<dx  +  5)  +  58. 

5.  When  3a:  -  12  >  35  -  bx,  and  4a;  -  12  >  6a:  -  31. 
Find  the  limits  of  x  and  y  in  the  following  case : 

6.  3a:  +  7«/  >  46, 

X  -  y  zzz  -1, 

181.  Type  Forms. — Inequalities  among  algebraic  quan- 
tities are  usually  established  by  reference  to  certain  stand- 
ard forms. 

The  following  is  a  very  important  standard  form : 
For  all  values  of  x  and  y  except  equality, 

x^  -\-  if  >  2xy.  (A) 

Proof. — {x  —  yY  is  essentially  positive  and  hence  >  0. 

.  •.     x^  -\-  y^  -  2xy  >  0. 

.'.     x^  -\-  y^  >  2xy. 

e.g.  The  sum  of  a  number  and  its  reciprocal  is  >  2. 
Let  X  denote  the  number.     Then  will 

a:+i  >  2. 

X 

Multiplying  both  members  by  x,  we  get 
a;2  +  1    >  2a:, 
or  a:^  +  1^  >  2a:  .  1. 

That  is,  the  first  inequality  is  true  if  the  last  is.  But 
we  know  that  the  last  is  true  by  reference  to  standard  (A). 
Hence  we  infer  that  the  first  is  also  true. 

Theorem  I.  The  product  of  two  positive  quantities 
whose  sum  is  constant  is  greatest  when  the  qua^itities  are 
equal 


274  INEQUALITIES. 

Denote  the  two  quantities  by  a  +  a:  and  a  —  x.  Then, 
whatever  value  be  assigned  to  x,  the  sum  of  the  quantities 
will  be  2«,  and  their  product  a^  —  x^.  Evidently  the 
product  will  be  greatest  when  a;  =  0;  that  is,  when  the 
quantities  are  equal. 

If  a  and  l  be  two  unequal  quantities,  the  two  halves  of 
their  sum  would  be  two  equal  quantities  whose  sum  would 
be  the  same  as  that  of  a  and  b.     Hence 

a  4-  h    a  4-  h         , 
la  +  JV        , 

^  >  ^^. 

.-.     a^h>  2Vab.  (B) 

Theorem  II.  The  product  of  any  number  of  positive 
quantities  ivhose  sum  is  constant  is  gi'eatest  when  the  quan- 
tities are  all  equal. 

For,  if  any  two  of  the  factors  are  unequal,  their  product 
would  be  increased  by  making  them  equal  without  chang- 
ing their  sum.  This  would  necessarily  increase  the  whole 
product  without  altering  the  sum  of  the  factors. 

If  a,  b,  c,  .  .  .  .  up  to  n  quantities  be  unequal,  by  tak- 
ing the  ^th  parts  of  their  sum  we  should  obtain  n  equal 
quantities  whose  sum  would  be  the  same  as  that  of  the  n 
unequal  quantities.     Hence 

(a-^b-\-c-\- .  .  .y 
or •  >  Vabc  .  .  . 


a-\-b-\-c-\- .  .  ,>  n  Vabc  .  .  ,  (0) 


INEQUALITIES.         .  276 

e.g.  a^-\-h'^>  2ai, 

and  .    a^-^b^-i-c^>  Ulc, 

in  all  cases  when  a,  h,  and  c  are  positive  and  unequal. 

EXERCISE  CXV. 
I. 

1.  For  what  value  of  x  would  16:^2  -f  25  =  40a:  ? 
Show  that  for  all  other  values  of  ^,  16a;^  -f  25  >  40a;. 

2.  Show  that  for  no  positive  integral  value  of  x  is 

x^-{-  —  <^x  -  —. 
5  5 

8.    Show  that  for  no  positive  value  of  a  can 

(3ff  +  2Z>)(3«  -  2^)  <  U{<oa  -  6b). 

4.     Show  that  (ab  +  xy){ax  -\-  by)  >  4:abxy. 

6.    Show  thsit  {b -]- c)(c -\- a) (a -{- b)  >  8abc, 

II. 

6.  If  a^-{-P=l,  and  x^-^y^=l,  show  that  ax-\-by<l. 

7.  If   a'^  -f  ^2  _[_  ^2  _  1^    ^Yi.d  x^  -\-  y^  -{-  z^  =  1,    show 
that  ax  -\-  by  -\-  cz  <  1. 

8.  Show  that  {x^y  -\-  yh  +  -2^^;)  {xy^-\-  yz^-\-  zx^)  >  9x^y^z^. 

9.  Show  that  «*  +  Z*''  >  rt^^  -|-  ab^,  except  when  a  and 
b  are  equal. 

10.  Show  that  a^  -\-  ¥  >  a^b  -\-  ab^,  except  when  a  and 
b  are  equal. 


CHAPTER  XXIII. 
RATIO  AND  PROPORTION. 

A.    EATIO. 

182.  Definition  of  Ratio. — The  term  ratio  denotes  the 
relation  which  one  quantity  bears  to  another  of  the  same 
kind  in  magnitude. 

The  magnitude  of  one  number  compared  with  another 
is  ascertained  by  dividing  the  number  by  the  one  with 
wliich  it  is  compared. 

When  the  number  is  a  multiple  of  the  one  with  which 
it  is  compared  its  ratio  to  it  may  be  expressed  by  an  inte- 
ger, otherwise  the  ratio  may  be  expressed  by  a  mixed  num- 
ber or  a  fraction. 

e.g.  The  ratio  of  12  to  4  =  12  ^  4  =  3;  the  ratio  of  3 
to  5  =  3  -4-  5  =  3/5;  the  ratio  of  13  to  4  =:  13/4  or  3^. 

The  ratio  of  one  number  to  another  might  be  defined 
as  the  number  by  which  the  second  must  be  multiplied  to 
produce  the  first. 

e.g.  5  must  be  multiplied  by  4  to  produce  20.  There- 
fore the  ratio  of  20  to  5  is  4. 

Again,  5  must  be  multiplied  by  3/5  to  produce  3. 
Therefore  the  ratio  of  3  to  5  is  3/5. 

183.  Expression  of  a  Ratio. — The  ratio  of  one  number 
to  a  second  may  be  expressed  either  by  writing  the  numbers 
in  the  form  of  a  fraction  with  the  first  number  as  the  nu- 
merator, or  by  writing  the  second  number  after  the  first 
with  a  colon  between. 


RATIO.  277 

e.g.  The  ratio  of  2  to  3  may  be  expressed  thus: 
|,     or     2:3. 

184.  The  Terms  of  a  Ratio. — The  first  term  of  a  ratio 
is  usually  called  the  antecedent,  and  the  second  term  the 
consequent. 

When  either  term  of  a  ratio  is  a  surd  the  ratio  cannot 
be  expressed  exactly  either  by  an  integer  or  by  a  rational 
fraction,  though  it  may  be  expressed  to  any  required  degree 
of  approximation,  by  carrying  out  the  extraction  of  the 
indicated  root  to  a  sufficient  number  of  places. 

e.g.  The  ratio  of  the  V'S  to  4  cannot  be  expressed 
exactly  by  any  rational  integer  or  fraction.     Thus, 

:^  =  ^^^:^««^  =.559017... 
4  4 

By  carrying  the  decimals  further  a  closer  approxima- 
tion may  be  obtained. 

185.  Kinds  of  Ratios. — When  the  antecedent  of  a  ratio 
is  equal  to  its  consequent,  the  value  of  the  ratio  is  one,  and 
the  ratio  is  said  to  be  a  ratio  of  equality ;  when  the  ante- 
cedent is  greater  than  the  consequent,  the  value  of  the  ratio 
is  greater  than  one,  and  the  ratio  is  said  to  be  a  ratio  of 
greater  inequality ;  and  when  the  antecedent  is  less  than 
the  consequent,  the  value  of  the  ratio  is  less  than  one,  and 
the  ratio  is  said  to  be  a  ratio  of  less  inequality. 


186.  Ratio  of  Equimultiples  and  Submultiples. — Since 
ma 

'  mV 
multiples 

Also, 

ratio  as  their  equi-submultiples,  equi-submultiples  being  the 


--——:,  two  numbers  have  the  same  ratio  as  their  equi 
0       mo  ^ 


Also,  since  t  —  ~n — '- — »  two  numbers   have  the  same 
0       0  ^  m 


278  RATIO  AND  PROPORTION. 

quotients  obtained  by  dividing  two  or  more  numbers  by  the 
same  number. 

187.  Theorem  I.  If  the  consequent  of  a  ratio  of 
greater  inequality  le  positive,  the  ratio  will  he  diminished 
hy  adding  the  same  positive  quantity  to  both  of  its  terms, 
and  increased  hy  subtracting  the  same  positive  quantity 
(less  than  the  conseque^U)  from  hoth  of  its  terms. 

Let  h  be  positive  and  a  >  h,  then  will  ,    ,       <  7-. 
^  h  -\-  X      h 


For 


a-{-  X      a       h(a-{-  x)  —  a(h  -|-  x)      x(h  —  a) 


h-^x      h  ~  h{h  +  x)  ~h{h  ^  x)' 

Now  since  a,  h,  and  x  are  positive  by  hypothesis  and 

0  <  a,  the  fraction  ,;-.    , — {  is  neffative.     .  •.    y— ' —  <  7-. 
h(h  ~\-  X)  ^  b  -\-  X      h 

.      .  a  —  X      a       x(a  —  b) 

But,  since  a  >  b,  a  —  b  is  positive,  and  since  x  <.b, 
b  —  X  is  positive. 


Hence  the  fraction  ~ !-  is  positive. 


h(b-  x)  ^"  ^ ^*    '  '  h-x^  h' 

188.  Theorem  II.  If  the  consequent  of  a  ratio  of  less 
inequality  he  positive,  the  ratio  will  be  increased  by  adding 
the  same  positive  quantity  to  both  of  its  terms,  and  dimin- 
ished by  subtracting  the  same  positive  quantity  (less  than 
the  consequent)  from  hoth  of  its  terms. 

a   I   X      a 
Let  h  be  positive  and  a  <  b,  then  will  ,    ,       >  7-,    and 

t)  -]-  X      0 

a  —  X      a 


b  —  X 

Prove  these  cases  in  the  same  manner  as  those  of  the 
last  section. 


PROPORTION.  279 

189.  Compound  Ratios.  —  When  the  antecedents  and 
also  the  consequents  of  two  or  more  ratios  are  multiplied 
together  the  ratios  are  said  to  be  compounded,  and  the  ratio 
of  the  products  is  called  the  compound  ratio  of  its  compo- 
nents. Thus,  ac  -.hd  is  the  compound  ratio  of  a  :  b  and 
c  :  d. 

When  a  ratio  is  compounded  with  itself  its  terms  are 
squared,  and  the  result  is  called  the  duplicate  ratio  of  the 
original.     Thus,  a^  :  b^  is  the  duplicate  of  a  :  b. 

Similarly  a^  :  b'^  is  called  the  triplicate  ratio  ot  a  :b. 

B.    PROPORTION". 

190.  Definition  of  Proportion. — Four  abstract  numbers 
are  said  to  he  2^ropo)'tionaL  or  to  form  2i  proportion^  when 
the  ratio  of  the  first  to  the  second  is  equal  to  that  of  the 
third  to  the  fourth.  Thus,  \i  a  :b  ^=  c  :  d,  the  four  quan- 
tities a,  b,  c,  and  d  form  a  proportion,  which  may  be 
written  in  any  one  of  the  following  ways: 

a       c 
a  -.b  —  c  -.d.     7-  =  -:,     or     a  \b  \\c  \d. 
b       d 

The  first  and  last  terms  of  a  proportion  are  called  the 
extremes,  and  the  second  and  third  terms,  the  means. 
Thus,  in  the  above  proportion  a  and  d  are  the  extremes, 
and  b  and  c  the  means. 

If  a,  b,  c,  d,  e,  etc.,  are  such  that  a  :b  —  b  :  c  =  c  :  d  = 
d  :  e,  then  a,  b,  c,  d,  e  are  said  to  be  in  continued  propor- 
tion. 

If  three  quantities,  a,  b,  c,  are  in  continued  proportion, 
so  that  a  '.b  =  b  :  c,  then  b  is  said  to  be  a  mean  proportional 
between  a  and  c. 

If  a  :  b  =  b  \  c  =  c  :  d,  then  b  and  c  are  said  to  be  two 
mean  proportionals  between  a  and  d,  and  so  on. 

191.  Test  of  the  Equality  of  Two  Ratios.  —  Since  a 


280  RATIO  AND  PROPORTION. 

ratio  is  virtually  a  fraction,  we  test  the  equality  of  two  ratios 
in  the  same  way  that  we  test  the  equality  of  two  fractions. 
Two  fractions  are  equal  if,  on  reduction  to  a  common 
denominator,  the  resulting  numerators  are  equal.     Thus, 

ft  c 

take  the  two  fractions  ^  and  -: ,  reduce  them  to  a  common 
0         a 

donominator,  and  we  have  -j-z  and    j^ .     These  resulting 

fractions  will  be  equal  when  ad  =  be.  Hence  the  four  quan- 
tities a,  b,  c,  d  are  proportional  when  the  product  of  the 
first  and  fourth  is  equal  to  the  product  of  the  second  and 
third ;  and,  conversely,  li  a  -.  h  =  c  :  d,  then  ad  =  he. 

In  any  p^^oportion  the  product  of  the  extremes  is  equal 
to  the  product  of  the  means.  This  is  the  great  numerical 
law  of  proportions. 

192.  Permutations  of  Proportions. — Any  interchange 
of  the  terms  of  a  proportion  is  permissible  which  does  not 
destroy  the  equality  of  the  product  of  the  extremes  and 
means.  The  various  interchanges  of  the  terms  of  a  pro- 
portion are  called  permutations. 

If  we  write  the  four  terms  of  a  proportion  in  the  four 
corners  formed  by  two  lines  which  cross  at  right  angles,  so 
that  the  first  ratio  shall  be  at  the  left  and  the  second  at  the 
right,  the  two  antecedents  will  be  at  the  top  and  the  two 
consequents  at  the  bottom,  and  the  extremes  will  be  in  one 
pair  of  opposite  corners  and  the  means  in  the  other.    Thus 

a     c 

in  the  form ,  «  :  J  is  the  first  ratio  and  c  :  d  the  sec- 

h     d 
ond ;  a  and  c  are  the  antecedents  and  h  and  d  the  conse- 
quents; a  and  d  are  the  extremes  and  h  and  c  the  means. 

The  letters  a  and  d  and  h  and  c,  which  stand  in  opposite 
corners  in  the  above  form,  may  be  called  the  opposites  of  a 
proportion,  and  we  may  make  the  general  statement  that 


pitopoBTioir. 


^81 


The  terms  of  a  proportion  may  he  ivritten  in  any  order, 
provided  the  opposites  remain  the  same. 

An  interchange  of  antecedent  and  consequent  in  each 
ratio  is  called  an  inversion,  an  interchange  of  an  antecedent 
of  one  ratio  with  the  consequent  of  another  is  called  an  al- 
ternation, and  an  interchange  of  one  ratio  with  another  a 
transpositioyi. 

There  are  seven  permutations  of  an  ordinary  proportion, 
so  that  when  four  quantities  are  proportional  they  may  be 
arranged  in  eight  different  ways. 

d 


Thus,  by  inversion 


— ,  and  by  mov- 

c 


C  1) 

—  becomes  — 

d  a 

ing  the  terms  of  each  of  these  successively  around  to  the 
right  eacli  of  the  above  may  be  changed  three  times  by 
alternation. 


Thus 


a 

c                      I 
—  becomes    — 

a 

d 

I                  c 
— ,     and    — 

d 

h 

d                    d 

c       c 

a                 a 

b 

d        .           a 
—  becomes  — 

h       c 

> 

a              d 
,  and 

c 

c 

c 

d 

d 

b 

b 

a 

And 


Write  out  in  the  ordinary  form  each  of  the  proportions 
given  above,  and  state  by  what  change  each  proportion  is 
obtained  from  the  last. 

193.  Transformations  of  Proportions.  —  Besides  these 
eight  permutations  there  are  other  transformations  which 
a  proportion  may  undergo. 

li  a  :b  —  c  '.d,  then  a  -\-  b  :  b  =  c  -\-  d  \  d. 


Let  —  =  X. 

0 


Therefore  a  =  bx. 


Then,  also,  ;t  =  ^-     (Why  ?)     Therefore  c  =  dx. 


282  RATIO  AND  PROPORTION, 

Then    — 7 —  becomes,  by  substitution, 

hx±h_h{x^l)_ 
—j~  -         ^—  _  a:  +  1. 

Also,  — ^ —  will  become j —  =  (x  4-  1). 

a  a  ^  ^ 

Therefore  — -, —  =  x  4-1  =  — -^ — . 

0  a 

Hence  a  -\~  b  :  b  =  c  -{-  d  :  d. 

This  change  is  called  composition. 

EXERCISE  CXVI. 

Prove  the  following  cases  by  methods  similar  to  the 
above : 

1.  a  —  b  :  b  =  c  —  d  :  d. 
This  change  is  called  division. 

2.  a  -{-  b  :  a  —  b  =  c  .-\-  d  :  c  —  d. 

This  change  is  called  composition  mid  division. 

3.  a  -\-  b  :  a  ^=^  c  -\-  d  :  c. 

4.  a  —  b\a  —  c  —  d\c. 

6.    If  a  -.b  —  c  '.  d  =  e  \f=  etc., 
then  a-{-c-{'e\b-\-d-\-f=a:b. 

This  change  is  called  addition. 

6.  li  a  '.b  =  c  \  d,  then  ma  :  mb  —  nc  :  nd. 

7.  Write  the  last  proportion  in  eight  different  ways. 

8.  \ia\b  =  c\d,  then  «"  :  ¥  -  c"  :  d"". 

9.  \t  a  '.b  ^=^  c  \  d,  and  m  '.n  ^=^  r  \  s, 
then  am  :  Z'm  =  cr  :  ds. 


PROPORTION.  283 

10.  li  a  :h  =^  c  \  d,  then 

la -\- mb  :  pa -{-  qb  =  Ic  -\-  md  :  pc  +  qd. 

11.  If  a  :h  —  c  :  d^  and  m  :  71  =  r  :  s,  then 

a  Vm  —  bVn:cVr  —  dVs  =  a  Vm  -\-  b  V71  :  c  Vr  -\-  d  Vs. 

EXERCISE  CXVII. 

Ex.  Which  is  the  greater  ratio,  a'^  -]-  b^  :  a  -\-  b  or 
a^  —  b^  :  a  —  b,  a  and  b  each  being  positive  ? 

Write  each  ratio  in  the  form  of  a  fraction,  and  subtract 
the  second  from  the  first,  and  show  that  the  result  is  essen- 
tially negative.  Hence  the  second  ratio  must  be  the  greater. 
Thus, 

a''-{-¥  _  a'-b'  _  {a'  +  b'){a  -  b)  -  {a'  -  b'){a  +  b) 
a-\-  b        a  —  b  ~  {a  -\-  b)(a  —  b) 

_    2ab'  -  ^a'b  . 
~  {a-^b){a-  b) 

"la^b  -  2ab' 


(a  +  b)(a  -  b) 

2ab{a^  -  b^) 
(a  +  b){a  -  b) 

2ab{a^  +  ab  +  b^) 
a^b 


Now  since  a  and  b  are  both  positive,  both  the  numera- 
tor and  the  denominator  of  this  fraction  must  be  positive. 

^4  _  54 
Hence    the  result  obtained    by  subtracting   7-  from 

^4  _|_    J4  ffi  _  ^4 

-—r-  is  negative.     Therefore 7  must  be  larger  than 

a-\-b  *^  a  —  b  ® 

a'  +  i' 
a-\-b' 


^84  RATIO  AND  PROPORTION. 


1.  Which  is  the  greater  ratio,  5  :  7  or  151  :  208  ? 

2.  Which  is  the  greater  ratio,  6  :  11  or  575  :  1056  ? 

3.  Which  is  the  greater  ratio,  7  :  12  or  589  :  1008  ? 

4.  Which  is  the  greater  ratio,  x^  -\-  y"^  :  x  -\-  y  or 
x^—  y'^  '.  X  —  y,  X  and  y  both  being  positive  ? 

5.  Which  is  the  greater  ratio,  x^  -\-  y^  :  x  -\-  y  or 
x^—  y^  '.  X  —  y,  X  and  y  both  being  positive? 

6.  Which  is  the  greater  ratio,  x"^  -\-  y'^  :  x  -{-  y  or 
x^  —  y""  :  X  —  y,  X  and  y  both  being  positive  ? 

7.  In  one  city  a  man  assessed  for  $10,000  pays  $72  tax, 
and  in  another  city  a  man  assessed  for  $720  pays  $4.50 
tax.     Compare  the  rate  of  taxation  in  the  two  cities. 

8.  Two  men  can  do  in  4  days  what  three  boys  can  do 
in  5  days.  Compare  a  man's  working  capacity  with  that  of 
a  boy. 

9.  For  what  vahie  of  x  will  the  ratio  b  -\-  x  -.  S  -{-  x 
become  5  :  8,  6  :  8,  7  :  8,  8  :  8,  9  :  8  ? 

10.  What  number  added  to  both  antecedent  and  con- 
sequent will  duplicate  the  ratio  3:4? 

n.  If  X  -^  1  \&  io  %(x  -{- 14)  m  the  duplicate  ratio  of 
5  :  8,  what  is  the  value  of  a;  ? 

II. 

12.  Find  two  numbers  in  the  ratio  of  7  :  12  such  that 
the  greater  exceeds  the  less  by  275. 

13.  What  number  must  be  added  to  each  term  of  the 
ratio  5  :  37  to  make  it  equal  to  1  :  3  ? 

14.  If  a;  :  y  =  3  :  4,  what  is  the  ratio  of  Ix  —  4y  \  3x 
+  73/? 


PROPORTION.  285 


16.     If  15(2a;2  —  y^)  =  "^xy^  what  is  the  ratio  oix  \  y'i 

16.  If  3(7a;2  _  24«/2)  =  _  29a;«/,  what  is  the  ratio  of 
x\y'i 

17.  What  is  the  least  integer  which  added  to  both 
terms  of  the  ratio  5  :  9  will  make  a  ratio  greater  than 
7  :  10? 

194.  Solution  of  Fractional  Equations. — When  an  equa- 
tion consists  of  two  fractions  only,  or  can  be  expressed  in 
the  form  of  two  fractions,  its  solution  may  be  simplified  by 
a  judicious  application  of  one  or  more  of  the  following 
principles  of  composition  and  division. 


Lel 

l\^%     Then 
h       d 

1°. 

a  —  c      a       c 
h-d^h~  d~ 

a  -{-  c 

-b-^d' 

2°. 

a-\-  b      c  -\-  d 
h     ~     d     ' 

3°. 

a  —  h      c  —  d 

h      ~     d    ' 

4° 

a-^h      c-\-d 

a  —  b      c  —  d' 

ft  Q 

Prove  the  first  of  these  cases  by  letting  —  =  ;,  =  ^'-  The 
remaining  three  have  already  been  proved. 

e.g.  1.    Solve  the  equation  — — -  =  — — -. 
•  X  -\-  ^     a-{-  b 

{x-4.)  +  {x^  4)  _  (^  -  5)  +  (^  +  5) 

(a;  -  4)  -  (a;  +  4)  ~  («  -  5)  -  («  +  5)' 

X  —  4:  -\-  X  ~\-  4:  _a  —  6  -\-  a  -\-  5 

ay  —  4:  —  X  —  4:  ~a  —  6  —  a  —  6' 

2x   _    2a 


Applying  4°, 


or 


or 


286  RATIO  AND  PROPORTION. 


or 


X  _a 
4  ~5' 

. '.     bx  =  4:a, 
e.g.  2.  Solve  the  equation 


Applying  1' 


or 


X  -\-  4:  —  b       X  -\-  4: 

{x-4:-\-b)-{x  —  4:)       X-  4: 

{x^4-I))-{x-\-  4)  ~  x-i-4:' 

b    _x  —  4: 
■=l"'^~+4* 

a:-4       -1 


X-\-4:  1 

Applying  4°,  .^  ^  _  =  o. 

.-.     x  =  0. 
e.g.  3.    Solve 

(x^2){x+5){x-\-3){x^S)  =  {x+l){x+Q){x+4:)(x+7). 

Dividing  both  sides  by  (x  +  3)(x  +  8)(^  +  4)(x  +  7), 
we  have 

(x-\-2)(x+5)  _{x-\-l)(x+e) 
{x  +  4.)(x+7)-{x+'S)(x-j-8y 

x^-\-7x-\-  10    _    x^-\-7x-\-6 
***     a;2  +  lla;  +  28    ~  x^ -{-  11a;  +  24* 

Applying  1°,  we  have 

(a;2  +  7a;  +  10)  -  {x^  -\- 7x -^  Q)    _  a:^  +  7a;  +  10 
(a;2  +  11a;  +  28)  -  {x^  +  11a;  +  24)  ~  a;^  +  11a;  -f-  28* 

4       arJ  +  7a;  +  10       1 


or 


4  ~  a;2  +  11a;  +  28  ~  1  • 

i«r«  +  7a;  +  10  =  a;2  _|.  113,  _|.  28, 


PROPORTION.  287 

or  —  4a;  =  18. 

.  •.  X  —  —  ^. 

e.g.  4.   Solve    [x  -  l)(2a;  -  3)^  =  {x  -  3)(2a;  -  1)1 
Dividing  both  sides  by  (2a;  —  3)2(2a;  —  1)^,  we  have 
a;  —  1  a;  —  3 


or 


or 


or 


7. 


(2a;  -  If  ~  (2a; 

-3)2' 

X  — 

1 

a; -3 

4a;2  -  4a;  +  1  ~  4a;2 

-  12a;  +  9' 

Applying 

1°,  we 

have 

{X 

-1)- 

{X  -  3) 

a;-l 

(4a;2-4a;  +  l)- 

(4a;2  -  12a;  +  9)       4a:2  _  4a;  +  1' 

2 

a;-  1 

1 

8(a 

^-1)" 

-  4:^2  _  4:^  _|_ 

■l-4(a:-l)- 

-• 

.      4(a; 

-  1)2  =  4a:2 

-  4a;  +  1, 

4a;2 

-  8a;  +  4  =: 

4a;2  _  4a;  +  1. 

.' 

•.      -  4a:  =  - 

-  3. 

.♦.     a:  =  3/4. 

EXERCISE  CXVIII. 

Solve  the 

following  equations 

X  —  a 

a      ~ 

h-  c 
c 

-I. 
2. 

x-b      l-l 

5      ~      7     * 

x-1   . 

x-\-l- 

1-  a 

4. 

a-  -  3      3  -  c 

a;  +  3  ~  3  +  c* 

2a;  +  3 
2.2:  -  3 

5 
"2" 

6. 

3a; -7       7 
3a:  +  7  ~  3  * 

mx  +  n 

&  + 

c  —  a 

^.           A 

3a;  +  4      c^a  —  l 

mx  —  n      6'  -|-  «  —  ^'         ■     3a;  —  4      a-\-l)  —  g 


288  RATIO  AND  PROPORTION. 


2x-\-l  1  3ic  -  1 

9.     ^  o    ,    J — r-T,  =  — r-T-     10. 


15. 


2a;2  +  2a;  +  3  "~  ic  +  1*  Sa:^  -  3^;  +  5      a;  -  1 

II. 

11.  (i»+l)(22;  +  5)2  =  4(a;  +  2)3. 

12.  (a:  -  l){x  -  %){x  +  6)  ^  (a;  +  2)(,r  --  3)(a:  +  4). 

13.  {x  -  l){x  -  ^y{x  -  5)  =  x{x  -  '6f{x  -  4). 

&x^  +  5x^  +  6a;  +  2    _  2a;^  4-  a;  H-  1 
^^-  6a;2  4-  5a;  +  3         ~      2./:  +  1      * 

W  -f  4a;^  +  8-^  +  4  _  3a;^  +  '^^  +  1 
9a;2  4-  4a;  +  5        ~"        3a;  +  2      * 

C.    VAKIATION. 

195.  Direct  Variation. — Suppose  x  and  y  to  represent 
two  variable  quantities  which  depend  upon  each  other  in 
such  a  way  that  when  one  changes  its  value,  the  other  must 
also  change  its  value;  and  let  x  and  y  be  so  related  that 
y  =  mx  (m  being  a  constant),  whatever  be  the  value  of  x; 
and  let  x^,  x^,  x^,  etc.,  and  ?/, ,  y^,  y^,  etc.,  be  corre- 
sponding values  of  x  and  y,  so  that  y^=  mx^ ,  y^=  nix^ ,  etc. 

Since  y  =  mx  and  y^  =  wa;,, 

y  _  mx   _  X 
y^      mx^      x^ ' 

Whence        y  :  y^=  x  -.  x^,     or    x  :  y  =  x^  :  y^. 

When  two  quantities  are  thus  related,  one  is  said  to 
vary  as  the  other.  Since  the  relation  is  mutual,  we  may 
say  that  y  varies  as  x,  or  that  x  varies  as  y.  The  symbol 
oc  denotes  this  relation,  and  is  read  "  varies  as  "  or  '^  varies 
directly  as."  Thus  y  a  a;  is  read  '' y  varies  as  a;";  and  x 
oc  y,  "  X  varies  as  y." 

To  say  that  y  varies  as  x  is  to  say  that  one  is  a  constant 
multiple  of  the  other,  or  that  they  so  vary  that  their  ratio 


VARIATION.  289 

remains  constant,  or  that  any  two  values  of  x  and  the  corre- 
sponding values  of  y  are  in  proportion. 

It  is  a  law  of  Optics  that  the  intensity  of  the  illumina- 
tion upon  a  surface  varies  directly  as  the  sine  of  the  angle 
which  the  rays  from  the  light  make  with  the  surface.  Tliat 
is,  the  larger  the  sine  of  this  angle,  or  the  more  nearly  per- 
pendicular the  rays  are  to  the  surface,  the  more  intense  is 
the  illumination.  If  two  surfaces  are  held  at  the  same  dis- 
tance from  the  light,  but  one  so  as  to  make  the  angle-sine 
for  the  rays  twice  as  great  as  for  the  other,  the  illumina- 
tion of  the  former  will  be  twice  as  intense  as  that  of  the 
latter;  if  the  surface  were  held  so  as  to  make  the  angle- 
sine  three  times  as  great,  the  illumination  would  be  three 
times  as  intense;  and  so  on.  While  the  illumination  in- 
creases with  the  size  of  the  angle,  it  does  not  increase  in 
the  same  ratio.  Hence  the  illumination  does  not  vary  as 
the  angle. 

196.  Inverse  Variation. — When  y  varies  as  x,  or  di- 
rectly as  X,  as  we  have  already  seen,  y  =  mx,  m  being  a 
constant. 

When  y  —  m— ,  y  is  said  to  vary  inversely  as  z.     That 

is,  y  increases  as  z  decreases,  and  vice  versa,  and  both 
change  at  the  same  rate. 

In  the  case  of  the  light,  the  intensity  of  the  illumina- 
tion on  a  surface  varies  with  the  distance  of  the  surface 
from  th^light,  the  intensity  becoming  less  as  the  distance 
becomes  greater,  and  the  intensity  changes  at  the  same  rate 
as  the  square  of  the  distance  changes.  Hence  we  say  that 
the  intensity  of  the  illumination  varies  inversely  as  the 
square  of  the  distance  from  the  light.  If  y  denote  the  in- 
tensity of  the  illumination,  z  the  distance  from  the  light, 
and  m  the  intensity  of  the  illumination  at  a  unit  distance 

from  the  source,  then  y  =  m-^ ,  and  y  cc  —. 

z  z 


290  EATIO  AND  PROrORTIOK 

1  X  .         . 

When  y  =  mx .  —  or  m  .  —,  y  varies  directly  as  x  and 

inversely  as  z.  In  the  case  of  the  light  already  considered, 
if  y  denote  the  intensity  of  the  illumination,  x  the  sine  of 
the  angle  which  the  rays  make  with  the  surface,  and  z  the 

X 

distance  from  the  light,  then  y—7n—^.   That  is,  the  intensity 

of  the  illumination  varies  directly  as  the  angle-sine  and 
inversely  as  the  square  of  the  distance. 

AVhen  y  =  mwx,  y  varies  jointly  as  w  and  x. 

If  to  denotes  the  intensity  of  the  source  of  light,  y  the 
intensity  of  the  illumination  on  the  surface,  x  the  angle- 

21)  X 

sine,  and  z  the  distance  from  the  source,  then  y  =  —5-. 

^         z^ 

Express  this  relation  in  words. 

197.  The  Constant  of  Variation. — In  all  the  cases  of 
variation,  the  constant  (m)  may  be  determined  when  any 
set  of  corresponding  values  is  given;  and  when  the  con- 
stant and  all  but  one  of  a  set  of  corresponding  values  are 
known,  the  remaining  one  can  be  calculated. 

e.g.  1.  A  ex  B,  and  when  A  =  8,  B  =  6.  What  will 
A  equal  when  B  =  24:? 

A  =  mB. 
,\  S  =  6m. 
.'.  m  =  3/4. 
.-.     ^  =  3/4x24  =  18. 

2.     ^  a  -^,  and  when  A  =  S,  B  =  Q.     What  will  A 

equal  when  ^  =  24  ? 

.  1 

A=m.^. 


VARIATION.  291 

,\     48  =  m. 

3.  A  (X  BC,  and  when  ^  =  2,  5  =  6  and  (7=4. 
What  will  A  equal  when  i?  =  18  and  C  =  6  ? 

A  =  m.  B.  C. 

2  =  m  X  6  X  4. 
.-.     m  =  1/12. 
.-.     A  =  1/12  X  18  X  6=  9. 

4.  A  cc  B .  -^,  and  when  ^  =  2,  B  =  6  and  C  =  4. 
What  will  ^  equal  when  B  =  18  and  C=Q? 

A  =  m  .  B  .  Yf. 

.-.     2  =  m.6.^. 
4 

.-.     m=4/.3. 

.-.     A  =4/3.  18.  1/6  =  4. 

EXERCISE    CXIX. 

I. 

1.  A  varies  as  B,  and  when  A  is  6,  5  is  4.  What  is  ^ 
when  5  is  9  ? 

2.  Jf  varies  inversely  as  iV,  and  when  if  is  4,  iV^is  13. 
What  is  M  when  iV^  is  20  ? 

3.  A  varies  as  B  and  C  jointly,  and  A  =  3  when  B  =  6 
and  (7=4.     What  is  ^  when  5  is  8  and  (7 is  3  ? 

4.  A  varies  as  ^  and  inversely  as  C,  and  A  =  4:  when 
5  =  6  and  (7  =  8.  What  is  the  value  of  A  when  B  =  IS 
and  (7  =  6  ? 


292  RATIO  AND  PROPORTION. 

5.  The  area  of  a  circle  varies  as  the  square  of  its  radius, 
and  the  area  of  a  circle  whose  radius  is  10  is  314.16.  What 
is  the  area  of  a  circle  whose  radius  is  20  ? 

II. 

6.  The  volume  of  a  sphere  varies  as  the  cube  of  its 
radius,  and  the  volume  of  a  sphere  whose  radius  is  1  foot  is 
4.188  cubic  feet.  What  is  the  volume  of  a  sphere  wliose 
radius  is  5  feet  ? 

7.  The  volume  of  a  cone  of  revolution  varies  as  its 
height  and  as  the  square  of  the  radius  of  its  base  jointly, 
and  the  volume  of  a  cone  7  feet  liigh  with  a  base  whose 
radius  is  3  feet  is  66  cubic  feet.  B'ind  the  volume  of  a  cone 
14  feet  high  with  a  base  whose  radius  is  18  feet. 

8.  The  volume  of  a  gas  varies  as  the  absolute  tempera- 
ture and  inversely  as  the  pressure  upon  it,  and  when  the 
temperature  is  280  and  the'  pressure  15  the  volume  of  a  cer- 
tain mass  of  a  gas  is  one  cubic  foot.  What  would  be  its 
volume  were  the  pressure  12  and  the  temperature  600  ? 

9.  The  distance  of  the  offing  at  sea  varies  as  the  square 
root  of  the  eye  above  sea-level,  and  the  distance  is  3  miles 
when  the  height  of  the  eye  is  6  feet.  What  is  the  distance 
when  the  height  is  72  yards  ? 

10.  The  intensity  of  illumination  varies  as  the  sine  of 
the  angle  which  the  rays  make  with  the  surface  and  in- 
versely as  the  square  of  the  distance  from  the  source,  and 
when  the  sine  and  distance  are  each  unity  the  illumination 
is  40.  What  will  be  the  illumination  when  the  sine  is  3/4 
and  the  distance  8  units  ? 


CHAPTER    XXIV. 
LOGARITHMS. 

198.  Definition  of  a  Logarithm.  —  In  the  expression 
cv^  zzz  y^  X  is  called  the  logarithm  of  y  to  the  base  a.  This 
relation  is  indicated  also  by  writing  x  —  log„  y. 

The  base  a  being  some  fixed  positive  number,  to  every 
value  of  y  there  is  a  corresponding  value  of  x,  and  to  every 
value  of  X  there  is  a  corresponding  value  of  y,  but  these 
values  are  often  incommensurable,  so  tliat  they  can  be  ex- 
pressed only  approximately. 

The  logarithm  of  a  number  may  be  defined  in  words  as 
the  index  of  the  power  to  which  a  given  base  must  he  raised 
to  jyroduce  the  number. 

A.       GENERAL    PROPERTIES    OF    LOGARITHMS. 

199.  The  Working  Rules  of  Logarithms. — 

Let  a"^  —  m.,    and     a-  —  n. 

Then  x  —  ^og^v:,     and     y  =  log^w. 

From  these  two  equations  we  may  deduce  four  impor- 
tant theorems : 

1°.  mn  =  r?^.  a^  =  cf^^', 

and  \og^{mn)  =  .t  +  ^; 

or  log„(w/0  =  loga^  +  logan- 

That  is,  the  logarithm  of  the  product  of  two  numMvs  is 
the  sum  of  the  logarithms  of  the  numbers. 

Of  course  this  theorem  may  be  extended  readily  to  the 
product  of  any  number  of  factors,  and  in  its  general  form 
it  would  be : 

293 


294  LOGARITHMS. 

The  logarithm  of  any  product  is  the  sum  of  the  loga- 
rithms of  its  factors. 

2°.  m-^  n  =  a''  -^  a^  =  a'^-y, 

and  loga(m  -i-  n)  =  x  —  y, 

or  loga(m  -^n)  =  log„m  -  log^n. 

That  is,  the  logarithm  of  the  quotient  of  two  numbers  is 
the  logarithm  of  the  dividend  minus  the  logarithm  of  the 
divisor. 

3°.  m^  =  {a^'Y  =  «^^, 

and  loga{m^)  =  px, 

or  \oga(m^)  =p\ogam. 

That  is,  the  logarithfn  of  a  power  of  a  number  is  the 
logarithm  of  the  number  multiplied  by  the  index  of  the 
power. 

and  log„(mV^)  =  1/p  .  x, 

or  log^(mV^)  =  l/p  log^m. 

That  is,  the  logarithm  of  a  root  of  a  number  is  the  loga- 
rithm of  the  number  divided  by  the  index  of  the  root. 

These  four  theorems  are  the  working  rules  of  logarithms 
as  applied  to  numhers. 

From  these  four  theorems  we  see  that  addition  of  loga- 
rithms corresponds  to  multiplication  of  numbers,  subtrac- 
tion of  logarithms  to  division  of  numbers,  the  multiplica- 
tion of  logarithms  by  numbers  to  the  raising  of  numbers  to 
powers,  and  the  division  of  logarithms  by  numbers  to  the 
extraction  of  roots  of  numbers.  There  are  no  operations 
on  logarithms  which  correspond  to  the  addition  and  sub- 
traction of  numbers,  and  there  is  no  operation  on  numbers 
in  ordinary  arithmetic  which  corresponds  to  the  raising  of 
logarithms  to  powers  or  to  the  extraction  of  their  roots. 


GENERAL  PROPERTIES  OF  LOOARITHMS.      295 

200.  Systems  of  Logarithms. — The  general  properties 
of  logarithms  are  the  same  for  all  bases,  and  any  positive 
number,  rational  or  irrational,  may  be  taken  as  a  base. 
Certain  numbers,  however,  otter  special  advantages  as  bases 
in  working  with  logarithms  and  in  calculating  them.  The 
base  which  is  most  advantageous  for  numerical  computation 
is  10,  and  the  one  most  advantageous  for  theoretical  inves- 
tigation is  the  incommensurable  2.7182818  ....  The  for- 
mer is  the  base  of  the  system  of  logarithms  in  common  use, 
and  the  latter  of  the  Napierean,  or  natural,  system  of  loga- 
rithms. 

201.  Common  Logarithms. — When  the  base  of  the  sys- 
tem is  10,  the  10  is  omitted  after  the  abbreviation  "log." 
Thus,  log  100  =  2,  means  that  10  must  be  raised  to  the 
second  power  to  produce  100.     Written  in  full  the  expres- 


sion  wo 

uld  be 

logiolOO  =  2. 

1  =  100, 

.  •.     log         1  =  0. 

10  =r  lOS 

.-.     log       10  =  1. 

100  ==  10^ 

.  •.     log     100  =  2. 

1000  =  103, 

.-.     log  1000  =  3. 

etc. 

Whenever  a  number  is  an  integral  power  of  ten,  its 
logarithm  is  a  positive  integer,  and  is  equal  to  one  less  than 
the  number  of  places  in  the  number  to  the  left  of  the  deci- 
mal point. 


••l-li^-^^-^ 

.-.      log       .1=-1. 

.01=1.,  =10- 

.-.     log    .01  =  -2. 

001  -  ^,  =  lo-^ 

.-.     log  .001  =  -3. 

0  =  .4,  =  10-% 

.  *.       loof.           0  =   —  00 

10 


296  LOGARITHMS. 

The  logarithm  of  0  is  negative  infinity.  Tlie  logarithm 
of  a  negative  number  is  imaginary.  Whenever  a  number 
is  a  decimal  and  equal  to  1  divided  by  an  integral  power  of 
10,  its  logarithm  is  a  negative  integer  and  is  equal  to  one 
more  than  the  zeros  to  the  right  of  the  decimal  point. 

Inasmuch  as  the  logarithm  of  any  number  to  base  10 
or  any  base  greater  than  1  increases  with  the  number,  it 
is  evident  from  the  above  that  the  logarithm  of  any  number 
greater  than  one  is  positive,  and  the  logarithm  of  any  num- 
less  than  one  is  negative;  also  that  the  logarithm  of  any 
number  between  1  and  10  lies  between  0  and  1,  and  is  a 
positive  decimal;  that  the  logarithm  of  any  number  be- 
tween 10  and  100  lies  between  1  and  2,  and  is  1  plus  a 
positive  decimal ;  and  so  on.  It  is  further  evident  that  the 
logarithm  of  any  number  between  1  and  .1  lies  between  0 
and  —  1,  and  is  —  1  plus  a  decimal;  that  the  logarithm  of 
any  number  between  .1  and  .01  lies  between  —  1  and  —  2, 
and  is  —  2  plus  a  decimal ;  and  so  on. 

202.  The  Characteristic  and  Mantissa  of  a  Logarithm. 

— In  general,  the  logarithm  of  a  number  is  composed  of 
two  parts,  an  integer  and  a  decimal.  The  decimal  part 
of  a  logarithm  is  incommensurable,  and  therefore  cannot 
be  expressed  exactly.  It  is  called  the  mantissa  of  the  loga- 
rithm, and  is  always  taken  as  positive. 

llie  integral  part  of  a  logarithm  is  positive  or  negative 
according  as  the  number  is  greater  or  less  than  one.  It  is 
called  the  characteristic  of  the  logarithm. 

The  method  of  computing  logarithms  cannot  be  con- 
sidered here.  Its  discussion  is  a  matter  of  Higher  Algebra. 
It  has  been  found  that 

6742  =  103'8276+^     . ..     log  6742  =  3.8276  +. 

Now         6-7420  =  6742  X  10  ==  lO^'S^^e  x  10^  =  W'^\ 

.'.     log  67420=  4.8276; 


GENERAL  PROPERTIES  OF  LOGARITHMS.      297 
and  674200  =  6742  X  100  =  lO^-^^^e  x  10^  =  io^-^r.\ 

.'.     log  674200  =  5.8276. 
Also,        674.2  =  6742  ^  10  =  lO^-^^^e  ^  iqi  =  io2.8276^ 

.-.     log  674.2  =  2.8276; 
and  67.42  =  6742  -f-  100  =  lO^-^s^e  ^  iq2  ^  ioi'8276^ 

..-.     log  67.42  =  1.8276, 
etc. 

We  see  from  the  above  that  so  long  as  the  figures  of  a 
number  and  their  arrangement  are  the  same,  the  mantissa 
of  the  logarithm  is  the  same  no  matter  what  position  the 
group  of  figures  may  occupy  in  the  scale  of  enumeration. 
The  shifting  of  the  group  of  figures  one  place  to  the  left 
increases  the  logarithm  by  unity,  because  it  multiplies  the 
number  by  10,  and  the  shifting  the  group  of  figures  one 
place  to  the  right  diminishes  the  logarithm  by  unity,  be- 
cause it  divides  the  number  by  10. 

This  property  of  logarithms  is  peculiar  to  the  system 
whose  base  is  10,  and  is  of  very  great  practical  importance. 

203.  Logarithmic  Tables. — The  mantissae  of  the  loga- 
ritlims  of  all  numbers  from  1  to  99999  have  been  calculated, 
and  published  in  the  form  of  tables.  In  these  tables  the 
approximation  in  the  mantissae  is  carried  sometimes  to  four, 
sometimes  to  five,  sometimes  to  six,  and  sometimes  to  seven 
decimal  places,  giving  rise  to  tables  of  four-place,  five-place, 
six-place,  and  seven-place  logarithms.  The  characteristics 
of  the  logarithms  are  not  given  in  these  tables,  because  these 
can  be  found  by  inspection  of  the  numbers. 

The  following  table  contains  the  mantissae  of  the  loga- 
rithms of  all  integers  from  100  to  1000,  calculated  to  four 
places  of  decimals,  and  from  it  can  be  found  approximately 
the  logarithms  of  all  numbers. 


298 


LOGARITHMS. 


COMMON    LOGARITHMS. 


GENERAL  PROPERTIES  OF  LOGARITHMS.      299 


commo:n^  logaeithms. 


n 

0 

1 

2 

3 

^ 

5 

6 

; 

8 

9 

d 

7 

60 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

61 
62 
63 

7R'53 
79-24 
7993 

7860 
7931 
8000 

7868 
7938 
8007 

7875 
7945 
8014 

7882 
7952 
8021 

7889 
7959 

8028 

7896 
7966 
8035 

7903 
7973 
8041 

7910 
7980 
8048 

7917 
7987 
8055 

6 

7 

64 
65 
66 

8062 
8129 
8195 

8069 
8180 
8202 

8075 
8142 
8209 

8082 
8149 
8215 

8089 
8156 
8222 

8096 
8162 
8228 

8102 
8169 
8235 

8109 
8176 
8241 

8116 
8182 
8248 

8122 
8189 
8254 

7 
6 

7 

67 

68 
69 

8261 
8325 
8388 

8267 
8331 
8395 

8274 
8338 
8401 

8280 
8344 
8407 

8287 
8351 
8414 

8293 
8357 
8420 

8299 
8363 
8426 

8306 
8370 
8432 

8312 
8376 
8439 

8319 
8382 
8445 

6 
6 
6 

7 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

71 
72 
73 

8513 
8573 
8633 

8519 
8579 
8639 

8525 
8585 
8645 

8531 
8591 
8651 

8537 
8597 
8657 

8543 
8603 
8663 

8549 
8609 
8669 

8555 
8615 
8675 

8561 
8621 
8681 

8567 
8627 
8686 

6 
6 
6 

74 
7o 
76 

8692 

8751 
8808 

8698 
8756 

8814 

8704 
8762 
8820 

8710 
8768 
8825 

8716 

8774 
8831 

8722 
8779 
8837 

8727 
8785 
8842 

8733 
8791 

8848 

8739 
8797 

8854 

8745 
8802 
8859 

6 
6 
6 

77 
78 
79 

8865 
8921 
8976 

8871 
8927 
8982 

8876 
8932 
8987 

8882 
8938 
8993 

8887 
8943 
8998 

8893 
8949 
9004 

8899 
8954 
9009 

8904 
8960 
9015 

8910 
8965 
9020 

8915 
8971 
9025 

6 
5 

6 

6 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

81 
82 
83 

9085 
9138 
9191 

9090 
9143 
9196 

9096 
9149 
9201 

9101 
9154 
9206 

9106 
9159 
9212 

9112 
9165 
9217 

9117 
9170 
9222 

9122 
9175 
9227 

9128 
9180 
9232 

9133 
9186 
9238 

5 
5 
5 

84 
85 
86 

9243 
9294 
9345 

9248 
9299 
9350 

9253 
9304 
9355 

9258 
9309 
9360 

9263 
9315 
9365 

9269 
9320 
9370 

9274 
9325 
9375 

9279 
9330 

9380 

9284 
9335 
9385 

9289 
9340 
9390 

5 

I 

87 
88 
89 

9395 
9445 
9494 

9400 
9450 
9499 

9405 
9455 
9504 

9410 
9460 
9509 

9415 
0465 
9513 

9420 
9469 
9518 

9425 
9474 
9523 

9430 
9479 
9528 

9435 
9484 
9533 

9440 
8489 
9538 

5 
5 
4 

4 

90 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

9581 

9586 

91 
92 
93 

9590 
96;38 
9685 

9595 
9643 
9689 

9600 
9647 
9694 

9605 

9652 
9699 

9609 
9657 
9703 

9614 
9661 
9708 

9619 
9666 
9713 

9624 
9671 
9717 

9628 
9675 
9722 

9633 
9680 
9727 

5 

5 
4 

94 
95 
96 

9731 
9777 
9823 

9736 
9782 
9827 

9741 
9786 
9832 

9745 
9791 
9836 

9750 
9795 
9841 

9754 
9800 
9845 

9759 
9805 
9850 

9763 
9809 
9854 

9768 
9814 
9859 

9773 
9818 
9863 

4 
5 
5 

97 
98 
99 

9868 
9912 
9956 

9872 
9917 
9961 

9877 
9921 
9965 

9S8] 
9926 
9969 

9886 
9930 
9974 

9890 
9934 

9978 

9894 
9939 
9983 

9899 
9943 
9987 

9903 
9948 
9991 

9908 
9952 
9996 

4 
4 
4 

204.  Method  of  Using  Logarithmic  Tables. — In  using 
a  table  of  logarithms  there  are  two  operations,  one  of  which 
is  the  inverse  of  the  other:  1°.  To  find  the  logarithm  of 
a  given  number;  2°.  To  find  the  number  which  has  a 
given  logarithm. 


300  LOGARITHMS. 

B.       TO    FIKD    THE   LOGARITHM    OF   A   NUMBER. 

1°.  Whe7i  the  Number  has  not  more  than  Three 
Figures. — First  determine  the  characteristic  by  inspection 
and  write  it  down.  Then  look  in  the  column  headed  n 
for  the  first  two  figures  of  the  number,  and  at  the  top  of 
the  columns  for  the  third  figure.  The  required  mantissa 
will  be  in  the  horizontal  line  of  the  first  two  figures  and  in 
the  column  which  has  the  third  figure  at  the  top.  This 
mantissa  should  be  written  after  the  characteristic  already 
found. 

e.g.     Find  the  logarithm  of  687. 

The  characteristic  is  2,  and  the  mantissa  found  in  the 
horizontal  line  of  68  in  the  left-hand  column  and  in  the 
column  of  7  at  the  top  is  8370.     Therefore 

log  687  =  2.8370. 

When  the  characteristic  is  negative,  the  minus  sign 
should  be  written  above  it,  to  indicate  that  it  is  the  charac- 
teristic alone  which  is  negative.  The  mantissse  of  the  tables 
are  always  positive.    Thus 

log  .0687  =  2.8370. 

When  the  number  consists  of  two  figures  only,  the  man- 
tissa is  found  in  the  column  headed  0.     Thus, 

log  68  =  1.8325. 

When  the  number  consists  of  one  figure  only,  consider 
the  second  figure  as  zero,  and  take  the  mantissa  from  the 
column  headed  0.  Thus  the  mantissa  of  6  is  found  in 
the  horizontal  line  of  60  in  the  column  headed  0. 


'in 

|0 


lo^  6  =  0.7782. 


2  .  When  the  Numl)er  has  more  than  Three  Figures. — 
When  a  number  has  more  than  three  figures,  use  must  be 
made  of  the  principle  that  when  the  difference  of  two  num- 


rO  FIND   THE  LOGARITHM  OF  A  NUMBEB.     301 

bers  is  small  compared  with  either  of  them,  these  differ- 
ences are  approximately  proportional  to  the  differences  of 
their  logarithms.  This  principle  is  called  the  Principle  of 
Proportional  Differences. 

e.g.     Find  the  logarithm  of  34567. 

log  34500  =  4.5378 
log  34600  =  4.5391 

Difference  of  the  mantissas  =         13 

Thus  a  difference  of  one  unit  in  the  third  place  corre- 
sponds to  a  difference  of  13  in  the  logarithms.  But  the 
given  number  differs  from  34500,  not  by  a  whole  unit  in  the 
third  place,  but  only  by  .67  of  that  unit.  Therefore  the  dif- 
ference between  the  logs  of  34500  and  34567  will  be  only 
.67  of  13  =  8.71,  which  we  take  as  9,  the  nearest  integer. 

Therefore  log  34567  =  4.5378 

9 


4.5387 

The  difference  between  one  mantissa  and  the  next  follow- 
ing in  the  tables  is  called  the  tabular  difference,  and  the 
result  obtained  by  multiplying  this  by  the  following  figures 
of  the  number  considered  as  a  decimal  is  called  the  real 
difference. 

It  is  never  necessary  to  use  more  than  three  of  the 
following  figures  for  a  multiplier,  and  seldom  more  than 
two. 

From  the  above  we  have  the  following  rule  for  finding 
the  logarithm  of  a  number  of  more  than  three  figures : 

Find  the  mantissa  of  the  first  three  figures  of  the  num- 
ber, and  the  tabular  difference. 

Multiply  this  tabular  difference  by  the  next  two  or  three 
figures  of  the  number,  considered  as  a  decimal,  and  add  the 
result  to  the  7nantissa  already  found. 


302 


LOGARITHMS. 


The  tabular  difference  should  be  taken  from  the  table 
at  sight.  To  facilitate  this  operation,  the  difference 
betweeii  the  last  mantissa  in  one  horizontal  line  and  the 
first  of  the  next  is  given  in  the  last  column,  headed  D. 

EXERCISE  CXX. 

Find  the  logarithms  of  the  following  numbers: 


1. 

956. 

2. 

58.7. 

3. 

2.38. 

4. 

.0325. 

5. 

50 

6. 

.003. 

7. 

40000. 

8. 

2. 

9. 

.000007. 

10. 

28645. 

11. 

16-.  327. 

12. 

.003579. 

13. 

2.468. 

14. 

8.006. 

C.       TO  FIND  A  KUMBER  WHICH   HAS  A  GIVEN  LOGARITHM. 

1°.  Whe7i  the  Exact  Mantissa  is  found  in  the  Tables. — 
Find  the  mantissa  in  the  table,  and  take  out  as  the  first 
two  figures  of  the  number  tlie  two  figures  of  the  column 
headed  N  which  are  on  the  horizontal  line  of  the  mantissa, 
and  as  the  third  figure  of  the  number  the  one  at  the  top  of 
the  column  in  which  the  mantissa  is  found,  and  point  off 
according  to  the  characteristic. 

e.g.     Find  the  number  whose  logarithm  is  1.9112. 

Find  9112  in  the  table  and  take  81  from  the  left-hand 
end  of  its  horizontal  line  and  5  from  the  top  of  its  column, 
and  place  the  decimal  point  before  the  8. 

log-^  1.9112  =  .815. 

The  symbol  log  ~  ^  means  the  7iumher  whose  log  is. 

2°.  When  the  Exact  Mantissa  is  not  found  in  the 
Table. — Take  out  from  the  table  the  next  smaller  mantissa, 
the  first  three  figures  of  the  corresponding  number,  and  the 
tabular  difference,  and  find  tlie  real  difference  between  this 


NUMBER  HAVING  A   GIVEN  LOGARITHM.        303 

mantissa  and  the  one  given.  Divide  the  real  difference  by 
the  tabular  difference  to  two  or,  at  most,  three  places  in  the 
quotient,  annex  these  figures  to  the  three  already  taken  out, 
and  point  off  accoi'ding  to  the  characteristic.  The  result 
is  seldom  trustworthy  to  even  two  places. 

It  will  be  seen  at  once  that  this  process  is  the  reverse 
of  that  for  finding  the  correction  of  the  mantissa  when  the 
number  has  more  than  three  figures. 

EXERCISE   CXXI. 

Find  the  numbers  which  have  the  following  logarithms*: 
1.     2.9355.  2.     f.5635.  3.     2.9948. 

4.     3.8845.  5.     0.5982.  6.     3.8340. 

7.     r.4570.  8.     2.9559.  9.     0.8077. 

205.  Cologarithms. — The  cologarithm  of  a  number  is 
the  logarithm  of  the  reciprocal  of  the  number. 

Thus,     colog  987  =  log  ^  =  log  1  -  log  987 

=  0-2.9943 

=  -2.9943. 

To  avoid  the  negative  mantissa,  the  logarithm  of  the 
number  is  usually  subtracted  from  10  instead  of  0. 

Thus,  colog  987  =  10  -  log  987, 

or  10  -  2.9943  =  .0057. 

Of  course  this  logarithm  is  10  too  large.  Such  a  loga- 
rithm is  called  an  augmenied  logarithm. 

The  colog  should  be  taken  from  the  table  at  sight.  We 
may  begin  at  the  left  hand  and  take  each  figure  from  9  till 
we  come  to  the  last,  which  should  be  taken  from  10. 


304  LOGARITHMS. 

EXERCISE  CXXII. 

Find  the  cologarithms  of  the  following  numbers : 
1.     3784.  2.     3959.  3.     2895. 

4.     .4265.  5.     .078976.  6.     .008. 

7.     50.  8.     .0008.  9.     .00009. 

D.       ARITHMETICAL   OPERATIONS. 

206.  Multiplication  by  Logarithms. — To  multiply  two 
or  more  factors  together  by  means  of  logarithms,  find  the 
logarithm  of  each  factor,  add  these  logarithms  and  then 
find  the  number  which  corresponds  to  this  resulting  loga- 
rithm. 

e.g.  Find  the  product  of  897,  564,  and  .0078. 

log  897  =  2.9528 
log  564=2.7513 
log  .0078  =  3.8921 


3.5962 
log -13.5962  =  3946.4 

207.  Division  by  Logarithms. — To  divide  one  factor 
by  another  by  means  of  logarithms,  find  the  logarithm 
of  each  factor,  subtract  the  logarithm  of  the  divisor  from 
that  of  the  dividend,  and  then  find  the  number  which  cor- 
responds to  the  logarithm  thus  obtained. 

As  in  many  practical  applications  it  is  necessary  to  per- 
form both  multiplication  and  division  in  the  same  example, 
it  is  preferable  in  all  cases  to  use  the  cologarithms  of  the 
factors  of  the  divisor  and  add  these  to  the  logarithms  of  the 
multiplication  factors. 

This  method  is  based  upon  the  principle  that  to  divide 
by  a  factor  is  equivalent  to  multiplying  by  its  reciprocal. 
In  using  cologarithms  it  must  be  borne  in  mind  that  each 


ARITHMEriGAL  OPERATIONS.  305 

colog  is  augmented,  and,  tlierefore,  that  as  many  lO's  must 
be  rejected  from  the  result  as  there  are  cologs  used. 

T..    .  ,.         1        »  526  X  862 
e.g.  Fmdthevalueof^3^— . 

log  526  =  2.7210 

log  862  =  2.9355 
colog  232  =  7.6345 
colog  683  =  7.1656 


20.4566 
log-^  0.4566  =  2.8613. 

208.  Involution  by  Logarithms. — To  raise  a  number 
to  a  power  by  means  of  logarithms,  find  the  logarithm  of 
the  number,  multiply  it  by  the  index  of  the  power,  and  find 
the  number  which  corresponds  to  the  resulting  logarithm. 

e.g.     Raise  249  to  the  sixth  power. 

log  (294)«  r=  2.4683  X  6 
=  14.8098. 
log-i  16.8098  =  645330000000000  approximately. 

209.  Evolution  by  Logarithms. — To  find  the  root  of  a 
number  by  means  of  logarithms,  take  out  the  logarithm  of 
the  number,  divide  it  by  the  index  of  the  root,  and  find  the 
number  which  corresponds  to  the  resulting  logarithm. 

If  the  characteristic  of  the  logarithm  is  negative,  before 
dividing  by  the  index,  add  as  many  tens  to  it  as  there  are 
units  in  the  index  of  the  root,  and  reject  ten  from  the  result- 
ing logarithm,  which  would  be  augmented  by  10.  For  this 
process  consists  in  adding  and  subtracting  the  same  multi- 
ple of  10  and  then  dividing  by  the  index  of  the  root. 

e.g.     Find  the  fifth  root  of  .086, 


306  LOGARITHMS. 

log  (.086)V5.z.  2.9345-^5 

=  (48.9345-50) -^  5 
=  (48.9345^5)- 10 
=1.7869. 

log~i  1.7869  =  .6121,  approximately. 

EXERCISE   CXXIII. 

Note. — A  negative  quantity  has  no  real  logarithm.  If 
such  quantities  occur  in  computation,  they  may  be  treated 
as  if  they  were  positive  and  then  the  sign  of  the  result  de- 
termined by  the  number  of  negative  factors.  If  this  num- 
ber be  even,  the  result  will  be  positive,  and  if  odd,  negative. 
In  arranging  the  logarithms  and  cologarithms  for  addition, 
it  is  best  to  place  an  n  after  each  one  which  has  been  found 
for  a  negative  factor,  and  then  a  glance  will  show  whether 
the  resulting  number  should  be  positive  or  negative. 

T.-   /.  ^1         1         .  23  X  -  8  X  -  6  ^ 
e.g.     J^md  the  value  of -^ . 

log  23  =  1.3617 

.  log    8  =  0.9031/i 
log    6  =  0.7782^ 
colog     5  =  9.3010 
colog  60  =  8.2218/i 


20.5658^ 
log-i  0.5658w=  -3.68. 
Find  by  logarithms  the  values  of  the  following: 
I. 
1.     250.42  X  .00687.  2.     -  7.8346  X  -  .086427. 

3,     -  9.896  X  12.857.  4.     .04632  X  .008764. 


ARITHMETICAL  OPERATIONS.  307 


5. 

.08 

7 

6. 

-  9.876 
.0076    • 

H 

18.009  X  -  .004 

8. 

27  X  -  82 

7. 

.007695  X  .004  * 

3.8  X  -  4.9* 

9. 

(86.42)3. 

10. 

(.0086)3. 

II. 

11. 

92/3. 

12. 

H. 

13. 

(-  3.278)^ 

14. 

192/3. 

15. 

(.12)«/5. 

(-  .000874)5/7. 

16. 
18. 

I^To: 

17. 

V'.  0009286. 

19. 

53/2  X  32/3, 

20. 

43/8 
5275- 

21. 

5643/5 

283   * 

22. 

^  5    *    3 

210.  Theorem.  The  logarithm  of  any  number  to  lase 
h  is  equal  to  the  product  of  the  logarithm  of  the  number  to 
the  base  a  by  logarithm  of  a  to  base  b. 

It  is  required  to  prove  logipi  —  log„m  .  logj,a. 

Let  logoW  =  Xj     and    log^m  =  y. 

Then  m  =  a?", 

and  m  —  y. 

Hence,  a  =  J^/*. 

And  .     a^/«  =  b. 


308  L00ABITHM8. 

and,  similarly,  logaO^  =  -. 

,\     logbfn  =  log„m  .  logi,a, 
or  ^gft^  _  log  ^,^, 

It  follows  from  the  above  theorem  that  if  the  logarithm 
of  any  number  to  base  b  is  known,  its  logarithm  to  any 
other  base  a  may  be  found  by  dividing  the  logarithm  of  the 
number  to  base  b  by  the  logarithm  of  a  to  base  b, 

e.g.     Find  log  3  to  base  7. 

Iogio3  =  0.4771. 
Iogio7  =  0.8451. 
0  4771 

EXERCISE  CXXIV. 

Find  the  following  logarithms : 
1.    logglS.  2.    log342.  3.    log^S. 

4.    Iog8.0803.  6.    Iogi5.007008.  6.    log956.31. 

When  the  number  can  be  expressed  as  an  exact  power 
of  the  base,  examples  like  the  above  may  be  solved  by  in- 
spection. 

e.g.     Find  the  value  of  logiel28. 

128  3=  16V*. 
.-.      log,el28  =  7/4. 
T,    log3729.  8.    log3,3125.  9.    loge,l/4. 


PART   II 
ELEMENTARY   SERIES 


I 


CHAPTER  XXV. 
VARIABLES  AND   LIMITS. 

211.  Constants  and  Variables. — A  number  which, 
under  the  conditions  of  the  problem  into  which  it  enters, 
may  assume  any  one  of  an  unlimited  number  of  values  is 
called  a  variable. 

A  number  which,  under  the  conditions  t)f  the  problem 
into  which  it  enters,  has  a  fixed  value  is  called  a  constant. 

Variables  are  usually  represented  by  the  last  letters,  x, 
2/,  z,  etc.,  of  the  alphabet,  and  constants  either  by  the  first 
letters,  a,  h,  c,  etc.,  or  by  Arabic  numerals. 

212.  Functions. — Two  variables  may  be  so  related  that 
a  change  in  the  value  of  one  produces  a  change  in  the  vah.e 
of  the  other.  In  this  case  one  variable  is  said  to  be  ^func- 
tion of  the  other. 

When  one  of  two  variables  is  a  function  of  the  other  tho 
relation  between  them  may  be  expressed  by  an  equation. 
Thus,  if  X  and  y  are  functions  of  each  other,  we  may  s:  y 

X  X 

that  —  =  a,  or  X  —  aii.  or  y  —  —. 
y  ''       ^      a     , 

Hence,  if  the  value  of  one  variable  be  assumed,  the 
corresponding  value  of  the  other  variable  may  be  computed. 
The  variable  for  which  values  are  assumed  is  called  Ihe  in- 
dependent  variable;  and  the  one  whose  value  is  found  by 
computation,  the  dependent  variable. 

When  an  equation  containing  two  variables  is  solved  for 
one  of  them,  the  variable  involved  in  the  answer  is  regarded 
as  the  independent  variable. 

811 


312  VARIABLES  AND  LIMITS. 

Thus,  in  equation  x  =  ay,  y  is  regarded  as  tlie  inde- 
pendent  variable ;  and  in  the  equation  y  =  -,  x  is  regarded 
as  the  independent  variable. 

213.  Limit  of  a  Variable. — As  a  variable  changes,  its 
value  may  approach  some  constant.  If  the  variable  can  be 
made  to  approach  a  constant  as  near  as  we  please  without 
ever  becoming  absolutely  equal  to  it,  the  constant  is  called 
the  limit  of  the  variable. 

214.  Axioms. — Any  quantity,  however  small,  may  be 
taken  times  enough  to  exceed  any  other  fixed  quantity, 
however  great. 

Conversely,  any  quantity,  however  great,  may  be  divided 
into  so  many  parts  that  each  part  shall  be  less  than  any 
other  fixed  quantity,  however  small. 

215.  Theorem  I.  If  a  fraction  have  a  finite  numer- 
ator and  an  independent  variable  for  its  denominator,  ive 
may  assign  to  this  denom/hiator  a  value  so  great  that  the 
value  of  the  fraction  shall  be  less  than  any  assignable  value. 

Let  a  be  the  numerator  of  the  fraction,  x  its  denomina- 
tor, and  c  any  finite  value,  however  small,  which  we  may 
choose  to  assign.  And  let  n  be  the  number  of  times  that 
we  must  take  c  to  make  it  greater  than  a.     Then 

a  <  nc. 
a 

.-.       -<6. 

n 
Hence,  by  taking  x  greater  than  7i,  we  shall  have 

-<c. 

x 

216.  Theorem  II.  If  a  fraction  have  a  finite  numer- 
ator and  an  independent  variable  for  its  denominator,  ice 


VARIABLES  AND  LIMITS.  313 

may  assign  to  this  denominator  a  value  so  small  that  the 
value  of  the  fraction  shall  exceed  any  assignable  value. 

Let  a  be  the  numerator  of  the  fraction,  x  its  denomi- 
nator, and  c  any  finite  value,  however  large,  which  we  may 
choose  to  assign. 

Let  ?i  be  a  number  greater  than  c.  Divide  a  into 
n  parts,  and  let  h  be  one  of  them.     Then 

a  =  nb. 

a 

Hence,  if  we  take  x  less  than  h, 
-  >  n>  c. 

X 

217.  Infinites. — If  a  variable  can  become  greater  than 
any  assigned  value,  however  great  that  value  may  be,  the 
variable  is  said  to  increase  indefinitely,  or  to  increase  with- 
out limit. 

When  a  variable  is  conceived  to  have  a  value  greater 
than  any  assigned  value  however  great,  the  variable  is 
said  to  become  infinite.  Such  a  variable  is  called  an 
infinite  number,  or  simply  an  infinite.  An  infinite  is 
usually  denoted  by  the  symbol  qo  . 

It  must  be  borne  in  mind  that  this  symbol  denotes,  not 
a  constant,  but  a  variable,  which  has  already  increased 
beyond  any  assignable  limit,  but  which  is  still  capable  of 
an  indefinite  increase. 

218.  Infinitesimals. — If  a  variable  can  become  less  than 
any  assignable  value,  however  small  that  value  may  be,  the 
variable  is  said  to  decrease  indefinitely,  or  decrease  without 
limit. 

In  this  case  the  variable  approaches  zero  as  a  limit. 

When  a  variable  which  approaches  zero  as  a  limit  is 
conceived  to  have  a  value  less  than  any  assigned  value, 


314  VARIABLES  AND  LIMITS. 

however  small  tliis  value  may  1)e,  the  variable  is  said  to 
become  infinitesimal.  Such  a  variable  is  called  an 
infinitesimal  number,  or  simply  an  infinitesimal.  An 
infinitesimal  is  often  denoted  by  the  symbol  0,  which 
in  this  case  must  be  understood  to  represent  an  exceed- 
ingly small  variable. 

We  often  express  the  relation  between  finite  quantities 
and  infinite  and  infinitesimal  quantities  as  follows : 

a  ^       n 

-  =  00  ,    —  =  0. 

0  00 


The  expression  tt  =  Qo  cannot  be  interpreted  literally, 
since  we  cannot  divide  by  absolute  0;  nor  can  the  expres- 
sion —  =  0  be  interpreted  literally,  since  we  cannot  find  a 

number  so  large  that  the  quotient  obtained  by  dividing  a  by 
it  shall  be  absolute  zero. 

The  expression  -  =  oo  is  simply  an  abbreviated  way  of 

writing:  when  x  approaches  zero  as  its  limit,  then  -  in- 
creases without  limit. 

—  =  0  is  simply  an  abbreviated  way  of  writing:  when 

X  increases  without  limit,  then  -  approaches  zero  as  its 
limit. 

219.  Approach  to  a  Limit.  —  When  a  variable  ap- 
proaches a  limit,  it  may  approach  it  in  one  of  three  ways : 

1°.  The  variable  may  be  always  less  tlian  its  limit; 

2°.  The  variable  may  be  always  greater  than  its  limit; 

3°.  The  variable  may  be  alternately  greater  and  less 
than  its  limit. 


VARIABLES  AND  LIMITS.  315 

If  X  represent  the  sum  of  w  terms  of  the  series 

■    '+\+\+\+---' 

X  is  always  less  than  its  limit  2. 

If  X  represent  the  sum  of  n  terms  of  the  series 

^      3-4-8 ' 

X  is  always  greater  than  its  limit  3. 

If  X  represent  the  sum  of  n  terms  of  the  series 

X  is  alternately  less  and  greater  than  its  limit  2. 

220.  Theorem  III.  If  k  he  any  fixed  quantity  how- 
ever great,  arid  x  he  a  variahle  which  ive  may  make  as 
small  as  we  please,  we  may  make  the  product  kx  less  than 
any  assignahle  quantity. 

If  there  be  any  smaller  value  of  kx,  let  it  be  denoted 
by  s.  Since  we  may  make  x  as  small  as  we  please,  let 
us  put 

.  *.     kx  <  s, 
so  that  s  cannot  be  the   smallest  value  of  the  product. 
Hence  the  product  cannot  have  a  smallest  value. 

221.  Theorem  IV.  If  two  functions  are  equal  they 
must  have  the  same  limit. 

Assume  it  possible  for  the  two  functions  to  have 
different  limits,  and  denote  these  limits  by  L  and  L '.     Put 

s  =  \(L-L'), 

SO  that  L  and  L '  differ  by  2*\ 

Now  since  L  is  the  limit  of  one  function,  that  function 
may  be  made  to  approach  L  so  as  to  differ  from  it  by  less 


316  VARIABLES  AND  LIMITS. 

than  A',  and  since  L'  is  the  limit  of  the  other  function,  this 
function  may  be  made  to  approach  L '  so  as  to  differ  from 
it  by  less  than  s.  And  as  the  difference  between  L  and 
L '  =  2s,  the  functions  in  the  above  case  must  be  unequal. 
But  this  is  contrary  to  the  hypothesis.  Hence  it  is  im- 
possible for  the  functions  to  have  different  limits. 

222.  Theorem  V.  The  limit  of  the  sum  of  several 
functions  is  equal  to  the  sum  of  their  separate  limits. 

Let  the  functions  be  denoted  by /(a:),  f{x'),  f{x"),  etc., 
and  their  limits  by  L,  L\  L  ",  etc. ;  and  let  the  differences 
from  their  limits  be  denoted  by  i,  i',  i'\  etc.     Then 

f{x)  =  L  —  i, 

f{x')=L'-^i\ 

f{x")  =  L"-i'\ 

etc.       etc. 

.-.    /(^) +/(,:')+/(:,")  + etc. 

=  i:  +  Z'  +  Z"  +  etc.  -  (i  +  i'  +  i"  +  etc.). 

We  must  now  prove  that  i  -\-  i'  -\-  i"  -[-  etc.  can  be  made 
less  than  any  quantity  we  can  assign. 

Let  h  denote  this  quantity,  which  may  be  as  small  as 
we  please; 
^i  denote  the  number  of  the  quantities  i,  %',  i", 
etc.; 
and        i  denote  the  largest  of  them. 

Since  the  difference  between  a  function  and  its  limit 
may  be  made  as  small  as  we  please,  we  may  make 

i  <  -,     or    ni  <  h. 

n 

But    i  -{-  i'  -\-  i"  -\-  etc.  <  ni^     (/  being  the  largest.) 
.*.     i  -\-  i'  -\-  i"  -\-  etc.  <  A. 


VARIABLES  AND  LIMITS.  317 

Therefore  L  -\-  L'  -\-  L"  -\-  etc.  is  the  limit  of 
/W+/(^')-f-/K)  +  etc. 

223.  Theoeem  VI.  The  limit  of  the  product  of  two 
functions  is  equal  to  the  product  of  their  separate  limits. 

Using  the  notation  of  Theorem  V,  we  have 

f(x)xf{x')  =  {L-i){L'-i-) 

=  L.  L'  -  (Li  -\-  L'i-W). 

Now  as  L  and  L '  are  finite,  Li'  -}-  L  i  can  be  made  as 
small  as  we  please,  and  therefore  the  quantity  within  the 
parenthesis  may  be  made  as  small  as  we  please.     Hence 

L.  L'  isthe  limit  of/(.T)  X /(«'). 

Cor.  1.  The  limit  of  the  product  of  any  numler  of 
functions  is  equal  to  the  product  of  their  limits. 

Cor.  2.  The  limit  of  any  power  of  a  function  is  equal 
to  the  power  of  its  limits  lohen  these  limits  are  not  both 
zero. 

224.  Theorem  VII.  The  limit  of  the  quotie^it  of  two 
functions  is  equal  to  the  quotient  of  their  limits  when  their 
limits  are  not  both  zero. 

Using  the  same  notation  as  before,  we  have 

f(x)         L  -  i 


f{x')-  L'-i" 

Now  the  difference  between  ^^-7   and  -^r-; — r,  is 

L  L  —t 

L'i-Li' 


L\L'-i')' 


The  numerator  of  this  expression  evidently  approaches 
zero  as  its  limit,  and  the  denominator  approaches  L  '^  as  its 
limit. 


318  VARIABLES  AND  LIMITS. 

Hence  the  expression  as  a  whole  has  zero  for  its  limit 
when  L '  is  not  itself  zero. 

226.  Definition. — The  expressions 


•^  ^  -"  X  —  a  J 


at 

denote  the  value  of  these  expressions  when  x  becomes  equal 
to  a. 

226.   Theoeem  VIII.     The  formula 


Lim 

X 


r  —  a  J  =  a"~ 


is  true  for  all  rational  values  of  n. 

Case  I.      When  n  is  a  positive  integer. 
We  have,  when  x  is  different  from  a, 

T-w   //" 

.M  -  1    _J_  n^n  -  8      I     ^2^n 


Now  suppose  X  to  approach  the  limit  a.  Then  :c"  ~  ^ 
will  approach  the  limit  6?"~\  x"'"^  tlie  limit  <3^"~2,  etc. 
Hence  «a:"~^  «^^a;"~*  etc.,  will  each  approach  the  limit 
^n  - 1  That  is,  each  term  of  the  second  member  approaches 
the  limit  a/"  ~  ^     Because  there  are  n  such  terms,  we  have 


r '  —  a' 
Lim.  


n  -  1 


X  —  a  J  =a 

Case  II.      When  n  is  a  positive  fractioti. 

Suppose  n  =  —,  p  and  q  being  whole  numbers.     Then 

a;^  —  a""      x'^  —  «« 


X  —  a  X  —  a 

Let  us  put,  for  convenience  in  writing, 


c^  =  y,         a"^  =  Jj 


VARIABLES  AND 

LIMITS. 

X 

=  r. 

a 

y- 

h" 

a;"-«" 

r- 

-h" 

- 1- 

■  h 

X  ~  a 

b" 

319 

then 

and 

X  ~  a         y*  —  0'^        I 

~y-o 

As  X  approaches  indefinitely  near  to  a,  and  consequently 
y  to  1),  the  numerator  of  this  fraction  (Case  I)  approaches 
to  pbP~^  as  its  limit,  and  the  denominator  to  qb'^'^.  Hence 
the  fraction  itself  approaches  to 

qb^-'         q  ' 

Substituting  for  b  its  value  «^  we  have 


X  —  a  J=a     q  q  (I 

Hence  the  same  formula  holds  when  n  is  a  positive 
fraction. 

Case  III.      Wlien  n  is  negative. 

Suppose  n  —  —  p,  p  itself  (without  the  minus  sign) 
being  supposed  positive.     Then 


x""  —  «"      X    ^  —  a 


p 


=  X- Pa- 


x—a 


fa''  -  x^\ 
\  X  —  a  I 

fxP  -  aP\ 
\  X  —  a  /' 


—  —  X  ^a 


When  X  approaches  a,  then  x'^  approaches  a'^,  and 

' approaches  ^«^  "^     Substituting  these  limiting 

values,  we  have 

I^im.  Elzi^  I      =  _  a-'^^pa^-'  =  - pa-^-K 
X  —  a  J=a 


VARIABLES  AND  LIMITS. 

Substituting  for  —  jt>  its  value  %  we  have 


Lim. =  na' 


Hence  the  formula 


X  —  a  J=a 


na' 


is  true  for  all  values  of  n,  whether  entire  or  fractional, 
positive  or  negative. 

227.  Definition  of  Series. — A  series  is  a  succession  of 
terms  formed  in  order  according  to  some  definite  law. 

228.  Theorem  IX.     The  limit  of  the  series 

A,  +  A,x  +  A^x'  +  A^x^  +  .  .  . 

when  X  is  i^idefinitely  diminished  is  A^,  provided  all  the 
coefficients  are  finite  and  the  coefficient  of  the  ntlt  term 
approaches  a  finite  limit  as  n  is  indefinitely  increased. 

1°.  Suppose  the  number  of  terms  of  the  series  to  be 
infinite. 

Let  Tc  denote  the  greatest  of  tlie  coefficients  A^,  A^, 
etc.,  and  denote  the  series  by  Aq  +  S, 

Since  Ic  is  the  largest  of  the  coefficients  A^,  A^,  etc. 

.  *.     hx  -\-  hx^  +  kx?  -f-  etc.  >  A^x  -\-  A^  -\-  A^x?  -j-  etc 
.  •.     ^^  <  Tex  +  lex?  +  hx?  +  etc. 

But  lex  -f-  Icx^  +  hx^  +  etc.  may  be  written  in  the  form 

lex 
of  the  fraction ,  as  may  be  shown  by  actual  division 

J.  —  X 

of  the  numerator  by  the  denominator. 


VARIABLES  AND  LIMITS.  321 

which,  when  x  is  indefinitely  diminished,  can  be  made  as 
small  as  we  please. 

Hence  by  indefinitely  diminishing  x,  Aq  can  be  made  to 
differ  from  the  series  by  less  than  any  assignable  quantity. 
Hence  Aq  becomes  the  limit  of  the  series. 

2°.  If  the  number  of  terms  in  the  series  is  finite, 
;S'  must  be  less  than  in  case  1°;  hence,  a  fortiori,  the 
theorem  is  true. 

229.  Theoeem  X.     In  the  series 

Ao  +  AiX  +  A^x^  +  A^x^  +  .  .  . 

by  taking  x  small  enough  ive  may  make  any  term  as  large 
as  ive  please  compared  with  the  sum.  of  all  that  folloiu  it, 
and,  hy  taking  x  large  enough,  we  can  make  any  term  as 
large  as  we  please  compared  with  the  sum  of  all  that 
precede  it. 

1°.  The  rth  term  of  the  series  will  be  A^^,  and  the 
ratio  of  this  to  the  sum  of  all  the  terms  that  follow  will  be 


AX 


A,^,x^^'-^A,^,x^^'-\-...        A,^ix-^A,^,x^-\-..: 

By  taking  x  small  enough  we  can  make  the  denominator 
of  this  last  fraction  as  small  as  we  please,  and  therefore  the 
fraction  itself  as  large  as  we  please. 

2°.  The  ratio  of  the  rth  term  to  the  sum  of  all  that 
precede  it  will  be 

Ajx''  A„ 


A,_,x-'^A,_X-'  +  ...  1  1  • 

""^x         ""^^  -r  •  '  ' 

By  taking  x  large  enough  we  may  make  the  denominator 
of  this  last  fraction  as  small  as  we  please,  and  therefore  the 
fraction  as  large  as  we  please. 


322  VARIABLES  AND  LIMITS. 

Cor.     In  an  expression  of  the  form 

consisting  of  a  finite  number  of  terms  in  descending  powers 
of  X,  by  taking  x  small  enough  we  may  disregard  all  the 
terms  but  the  last,  and  by  taking  x  large  enough  we  may 
disregard  all  the  terms  but  the  first. 

230.  Vanishing  Fractions. — A  fraction  which  assumes 

tlie  form  —  for  some  particular  value  of  x  is  called  a  van- 

ialiing  fractioyi. 

The  fraction,  though  indeterminate  in  form  when  x  has 
this  critical  value,  has  a  real  value.  To  determine  this 
value  is  to  evaluate  the  fraction. 

Sometimes   for   a   particular   value   of   x  the  fraction 

assumes  the  form       ,  which  is  also  indeterminate  in  form. 

oo 

The  values  of  the  fractions  when  they  assume  these 
indeterminate  forms  are  really  the  limiting  values  of  the 
fractions  as  x  is  indefinitely  increased  or  diminished. 

The  limiting  value  of  a  fraction  when  x  in  both 
numerator  and  denominator  is  indefinitely  increased  or 
diminished  may  be  found  by  Theorem  X,  cor. 

e.g.     Find  the  limiting  value  of  ,,  ^   ,   ^\       ,  when  x 

dx-'  -\-  7x^  —  4 

is  infinite  and  when  x  is  zero. 

1°.  When  a;  =  00 ,  every  term  except  the  first  of  the 
numerator  may  be  disregarded,  and  we  have  as  the  limiting 
value 

4:X^         4 

Sx^^S' 

2°.  When  x  =  0,  every  term  except   the  last  of  the 

numerator  and  denominator  may  be  disregarded,  and  we 

7 
have  as  the  limiting  value  —  -. 


VARIABLES  AND  LIMITS.  323 

The  limiting  value  of  a  fraction  which  assumes  an 
indeterminate  form  for  a  critical  value  of  x  may  be  found 
by  first  removing  from  the  numerator  and  denominator  all 
common  factors  in  x,  and  substituting  the  critical  value  of 
X  in  the  result. 

e.g.     Find  the  limiting  value  of 

x^  —  4:ax  -j-  'Sa^ 


x^-aJ' 

3n  X  =  a. 

X?  -  4:ax  +  da  _ix-  a)(x  -  3a)  _ 
x^  —  a^         ~  {x  —  a)(x  -\-  a)  ~ 

x  —  da 

X  -\-  a' 

Put  x  =  am  this  result,  and  we  have 

^^=       1. 
2« 

EXERCISE  CXXV. 

Find  the  limiting  values  of  the  following: 

X  —  al  X  —  a~] 

2.     

«      J^ao.  X     J=o. 

ax  -\-b~\  ax  -\-h'l 

bx  -\-  aJ  =  ao'  '     bx-^  aj=  0- 

mx^     ~[  mx^     "I 

2)01?  —  ax  J  =  00 .  '    p^  —  ax  J  _  q. 

{2x  -  3)(S  -  6x)-\  (2x  -  3) (3  -  da;)"] 
7x^-Qx-{-4:    J  =00.      ®"         7x^-6x-{-4:    J.. 

x^  —  a^~]  afi  —  z^~] 

10.     

X  —  a  J=a'   ■  X  —  z    J^  g. 


0. 


11. 


+  1~]  x^-8x-{- 15-| 


324  VABIABLES  AND  LIMITS. 

231.  Discussion  of  Problems. — To  discuss  the  solution 
of  a  problem  when  the  answer  is  literal  is  to  observe 
between  what  limiting  numerical  values  of  the  known 
elements  the  problem  is  possible,  and  whether  any  singu- 
larities or  remarkable  circumstances  occur  within  these 
limits. 

The  following  discussions  will  serve  to  illustrate  tlie 
significance  of  indeterminate  forms  of  expression,  and  of  0 
and  00  .as  limiting  values. 

a.  The  Product  of  Two  Quantities  whose  Sum  is  Constant. 

Divide  a  into  two  parts  whose  product  shall  equal  h. 
Let  X  and  y   denote   the  parts.     Then,  by  the  con- 
ditions. 


x-\-  y  =  a. 

(1) 

xy  =  b. 

(2) 

From  (1),                        y  —  a  —  X. 

By  substitution  in  (2), 

x{a  —  .t)  =  l. 

.-.     x^  -  ax^h^^', 

3nce                           x=-a±^--l). 

1             /  a^ 
and  the  two  parts  are  o"  ^*^  +\/  ^: ^' 

and  VsJt-^- 


VARIABLES  AND  LIMITS.  ^25 

Now  these  values  are  imaginary  \f  h  >  j- ;  that  is,  if  the 

product  of  the  two  parts  is  greater  than  the  square  of  half 
their  sum. 

Cor.  Tlie  product  of  tiuo  quantities  cannot  he  greater 
than  the  square  of  half  their  sum. 

Or,  the  product  of  tivo  parts  of  a  given  quantitij  is 
qreatest  when  those  parts  are  equal. 

The  two  parts  will  be  incommensurable  when  the  differ- 
ence between  their  product  and  the  square  of  half  their 
sum  is  not  a  perfect  square. 

b.    The  General  Quadratic  Equation. 
The  equation 

ax^  -^Ix-^-  c  —  0 

has  been  discussed  already  in  so  far  as  to  observe  when  the 
values  of  x  become  imaginary,  when  they  are  real  and 
rational,  when  real  and  irrational,  and  when  equal. 

We  will  now  discuss  some  peculiarities  which  may  arise 
by  the  vanishing  of  each  of  the  coefficients  in  turn. 

Note  that  c  is  really  the  coefficient  of  x^. 

If  c  =  0,  then 

ax^-{-hx  =  0',  (1) 

whence  x  =  0,    or . 

a 

That  is,  one  of  the  roots  is  zero  and  the  other  is  finite. 
If  ^>  =  0,  then 

rf_|_c^O;  (2) 


whence 


=v-^ 


326  VARIABLES  AND  LIMITS. 

In  this  case  the  roots  are  equal  in  vahie  and  opposite  in 
sign. 

They  will  be  real  or  imaginary  according  as  a  and  c 
have  opposite  signs  or  tlie  same  sign. 

It  a  =  0,  then 

hx  +  c-Q',  (3) 

and  apparently  in  this  case  the  quadratic  has  but  one  root. 

namely,  —  — .     But  every  quadratic  equation  has  two  roots, 

and  in  order  to  discuss  the  values  of  these  roots  we  may 
proceed  as  follows : 

Put  -  for  X  in  the  original  equation,  and  clear  of  frac- 

y 

tions.     Then 


of 

+  %  + 

a  =  0. 

Now 

put  a 

=  0, 

and 

we  have 

oy^ 

^hy  = 

0; 

mce 

2/  = 

=  0,    or 

_5 
c' 

••• 

X  - 

1 

1 
c 

= 

=  00 ,    or 

c 

Hence,  in  any  quadratic  equation  one  root  becomes 
infinite  when  the  coefficient  of  x^  becomes  zero. 

This  is  merely  a  convenient  abbreviation  of  the  follow- 
ing fuller  statement : 

In  the  equation  ax^  -{-  hx  -\-  c  =  0,  if  a  is  very  small 


VARIABLES  AND  LIMITS.  327 

one  root  is  very  large,  and  becomes  indefinitely  great  as  a 
is  indefinitely  diminished.      In    this  case  the  finite  root 

approaches  —  —  as  its  limit. 

c.    The  ProUem  of  the  Houriers. 

Two  couriers,  A  and  B,  are  travelling  along  the  same 
road  in  the  same  direction,  BR',  at  the  respective  rates  of 
m  and  w  miles  an  hour.  At  a  given  hour  A  is  at  P,  and 
B  is  a  miles  beyond  him  at  Q.  After  how  many  hours, 
and  how  many  miles  beyond  P,  will  the  couriers  be 
together  ? 

R ^ 9. .R' 

Let  X  denote  the  number  of  hours  after  the  given  time, 
and  y  the  number  of  miles  beyond  P.     Then 

y  —  a  —  number  of  miles  beyond  §; 

y=mx',  (1) 

y  —  a=  nx.  (2) 

From  (1)  and  (2), 

a  ,  am 

X  = — ,    and    y  = . 

7n  —  n  *^      m  —  n 

I. 
Suppose  a  to  be  positive. 
1°.  Let  m  >  n. 

In  this  case  both  x  and  y  will  be  positive,  and  A  will 
overtake  B  to  the  right  of  P. 

This  corresponds  with  the  hypothesis;  for  since  a  is 
positive,  B  is  ahead  of  A,  and  since  m  is  greater  than  w, 
A  is  travelling  faster  than  B. 


328  VARIABLES  AND  LIMITS. 

2°.     Let  m  =  71. 

In  this  case  the  values  of  x  and  ii  take  —   and    -— ,  and 

each  becomes  00 . 

This  result  indicates  that  one  never  would  overtake  the 
other. 

This  interpretation  corresponds  with  the  hypothesis 
made.  For  B  is  «  miles  ahead  of  A,  and  both  are  travelling 
at  the  same  rate. 

3°.     Let  m.  <  71. 

In  this  case  the  values  of  x  and  y  both  become  nega- 
tive. This  indicates  that  che  couriers  were  together  before 
the  given  time  and  before  they  reached  the  point  P. 

This  corresponds  with  the  supposition;  for  B  travels 
faster  and  is  ahead  of  A  at  the  given  time.  He  therefore 
must  have  overtaken  A  and  have  passed  him  before  the 
given  time. 


II. 


Suppose  «  =  0. 
1°.     Let  771  >  n. 

In  this  case  the  values  of  x  and  y  both  assume  the  form 
0 


=  0. 
m  —  71 

This  is  as  it  should  be;  for  since  the  couriers  travel  at 
unequal  rates  and  are  together  at  the  given  hour,  they 
never  could  have  been  together  before,  nor  can  they  be  to- 
gether again  afterward.  As  A  travels  faster  than  B,  he 
must  have  overtaken  B  just  at  the  given  time. 

2°.     Let  771  =  71. 

In  this  case  the  values  of  x  and  y  both  assume  the  form 

-,  and  the  problem  becomes  indeterminate. 


VARIABLES  AND  LIMITS  329 

This  corresponds  with  the  given  conditions;  for  the 
couriers  are  together  and  travelling  at  the  same  rate. 
Hence  they  must  have  been  together  during  all  their  past 
journey,  and  they  must  continue  together  for  the  future. 

3°.     Let  m  <  n. 

This  gives  the  same  results  as  1°,  the  only  difference 
being  that  B  must  have  overtaken  A  at  the  given  time. 

III. 

Suppose  a  to  be  negative. 

1°.     Let  m  >  71. 

In  this  case  x  and  y  are  both  negative,  and  the  couriers 
must  have  been  together  on  the  road  some  time  before  the 
given  hour. 

This  corresponds  with  the  supposition ;  for  A,  being  now 
ahead  and  travelling  faster,  must  have  passed  B  at  some 
previous  point. 

2°.     Let  m  =  n. 

This  will  again  give  oo  for  both  x  and  y,  and  the  prob- 
lem is  impossible. 

These  results  evidently  suit  the  conditions  of  the  prob- 
lem; for  A  is  now  ahead,  and  both  are  travelling  at  the 
same  rate.  Hence  the  couriers  never  could  have  been  to- 
gether in  the  past,  and  never  can  be  in  the  future. 

3°.     Let  m  <  n. 

In  this  case  x  and  y  must  both  be  positive,  and  the 
couriers  must  be  together  at  some  point  farther  along  the 
road. 

This  also  answers  to  the  given  conditions;  for  B  is  now 
behind  at  the  given  time,  and  travelling  faster.  Hence  he 
must  overtake  A  at  some  future  point. 


330  VARIABLES  AND  LIMITS. 

d.    The  ProUem  of  the  Lights. 

Two  lights,  A  and  B,  of  given  intensities,  are  situated 
at  a  given  distance  apart.  Find  the  point  on  the  line  AB 
where  the  lights  give  equal  illumination. 

Let  m  =  illumination  of  A  at  a  unit's  distance, 

a  =  distance  from  A  to  B, 
and         X  —  distance  from  A  to  P,  the  point  of  equal  illu- 
mination. 
Then  a  —  x  will  be  the  distance  from  B  to  P. 
Since  the  illumination  at  P  varies  directly  as  the  inten- 
sity of  the  source  and  inversely  as  the  square  of  its  distance, 

the   illumination   of  A  at  P  will  be  -g,  and  of  B  at  P 


{a  -  xy 

By  hypothesis  these  two  illuminations  are  to  be  equal. 


m 

= 

n 

x' 

{a 

-xf 

/>• 

a  Via 

Whence 

Vm  ±  Vn 

The  double  sign  of  the  denominator  gives  two  values  for 
X,  and  shows  that  there  must  be  two  points  of  equal  illumi- 
nation. 

I. 

Suppose  a  to  be  positive. 
1°.     Let  m  >  71. 

In  this  case  both  values  of  x  will  be  positive,  one  less 
and  the  other  greater  than  a,  and  the  one  which  is  less 

than  a  will  be  greater  than  — ,  since  the  denominator  of  the 


VARIABLES  AND  LIMITS.  331 

fraction  is  less  than  2  Vm.  Hence  the  two  points  of  equal 
illumination  will  both  be  on  the  same  side  of  A,  one  be- 
tween A  and  B  and  the  other  beyond  B ;  and  the  one  be- 
tween A  and  B  will  be  nearer  to  B  than  to  A. 

Evidently  these  results  are  what  we  ought  to  expect. 
The  point  of  equal  illumination  between  the  lights  ought 
to  be  nearer  the  less  intense  light,  and  the  second  point  of 
illumination  ought  to  be  beyond  the  less  intense  light,  so  as 
to  be  nearer  to  it  than  to  the  more  intense  light. 

2°.     Let  m  =  n. 

In  this  case  the  first  value  of  x  will  be  positive  and 

equal  to  — ,  and  the  second  value  of  x  will  be  oo  . 

That  is,  one  of  the  points  of  equal  illumination  will  be 
midway  between  the  lights,  and  the  other  must  be  at  in- 
finity. 

The  lights  being  of  equal  intensity,  the  points  of  equal 
illumination  ought  to  be  equally  distant  from  them,  and 
the  only  such  points  are  the  one  half  way  between  the  two 
lights  and  the  point  at  infinity,  or  nowhere. 

3°.     Let  m  <  n. 

In  this  case  the  first  value  of  ./•  will  be  positive  and  less 

than  — ,  and  the  second  value  will  be  negative  and  greater 

Z 

than  a. 

That  is,  one  of  the  points  of  equal  illumination  will  be 
between  A  and  B  and  nearer  the  less  intense  light,  and  the 
other  is  on.  the  opposite  side  of  A  to  B,  so  as  also  to  be 
nearer  the  less  intense  light,  A. 

II. 

Suppose  a  to  be  zero. 
1°.     Let  m  >  n. 

In  this  case  both  values  of  x  become  zero,  and  both 
illuminations  become  co . 


332  VARIABLES  AND  LIMITS. 

These  results  are  on  the  supposition  that  each  light  is  a 
mathematical  point,  which  is  physically  impossible. 

Mathematical  analysis  does  not  concern  itself  with  phy- 
sical impossibilities.  Could  each  light  be  reduced  to  a 
mathematical  point,  the  intensity  of  the  light  would  become 
infinite  at  that  point,  and  were  the  two  lights  together  at 
that  point,  both  illuminations  would  be  equal  there  and 
nowhere  else. 

3°.     Let  m  <n. 

The  result  in  this  case  would  be  the  same  as  in  1°. 

III. 

Suppose  a  to  be  negative. 

The  student  may  discuss  this  case  when  m  y  71,  m  =  n, 
and  m  <  n.  The  conclusions  will  be  similar  to  those  of  i, 
though  not  identically  the  same. 


CHAPTER  XXVI. 
THE  PROGRESSIONS. 

A.       ARITHMETICAL    PROGRESSION. 

232.  Arithmetical  Series. — When  tlie  terms  of  a  series 
increase  or  decrease  by  a  common  difference,  it  is  called  an 
aritJmietical  series  or  an  arithmetical  progression.  This 
series  is  denoted  by  the  letters  A.  P. 

Each  of  the  following  series  represents  an  arithmetical 
progression : 

1,  4,  7,  10,  etc. 

3,  -  1,  -  5,  -  9,  etc. 

a  —  4c?,  a  —  d,  a  -^  2d,  etc. 

In  the  iirst,  the  common  difference  is  3 ;  in  the  second, 
—  4;  and  in  the  third,  3d. 

The  general  type  of  an  A.  P.  is 

a,  a  -\-  d,  a  -\-  2d,  a  -{-  3d,  etc., 

in  which  a  is  the  first  term,  and  d  the  common  difference. 

233.  The   nth   Term  of  an  Arithmetical  Progression. 

— Observe  that  the  coefficient  of  d  in  any  term  of  the  type 
is  one  less  than  the  number  of  the  term,  it  being  1  in  the 
second  term,  2  in  the  third  term,  3  in  the  fourth  term,  etc. 
Hence  the  nth  term  of  an  arithmetical  progression  will 
be 

a-\-  {n  —  l)d. 

333 


334  THE  PROGRESSIONS. 

Thus  the  fifteenth  term  of  an  arithmetical  progression 
whose  first  term  is  5  and  whose  common  difference  is  3 
will  be 

5  +  (15  -  1)3  =  47. 

When  any  two  terms  of  an  arithmetical  progression  are 
given,  the  common  difference,  and  any  other  term,  may 
be  found  by  the  formula  for  the  nih.  term. 

e.g.  Suppose  the  twelfth  term  of  an  arithmetical  pro- 
gression to  be  36,  and  the  eighteenth  term  to  be  12.  Find 
the  first  term,  the  common  difference,  and  the  sixth  term 
of  the  progression. 

Let  a  denote  the  first  term,  and  d  the  common  differ- 
ence. 

The  twelfth  term  will  \)q  a  -\-  lid,  and  the  eighteenth 
term,  a  -\-lld. 

.-.     «+17^=12, 

and  a  +  11^/  =  36. 

6fi=-24 

and  ^/  =  —  4. 

Also,  «  =  36  -  11(-  4)  =  80. 

Therefore  the  sixth  term  will  be 

80  +  5(-  4)  =  GO. 

234.  Arithmetical  Means. — When  three  quantities  are 
in  arithmetical  progression,  the  second  is  called  the  arith- 
metical mean  of  the  other  two. 

Thus,  if  «,  h,  and  c  are  in  A. P.,  h  is  the  arithmetical 
mean  of  a  and  c. 

By  definition,         h  —  a  —  c  —  h, 

or  2^  =  «  +  c. 

h  =  l/2(«  +  c). 


ARITHMETICAL  PB00RES8I0K  335 

Hence,  tlie  arithmetical  mean  of  two  quantities  is  half 
their  sum. 

When  any  number  of  quantities  are  in  arithmetical  pro- 
gression, all  the  intermediate  terms  are  called  arithmetic 
means  of  the  two  extreme  terms. 

Any  number  of  arithmetical  means  may  be  inserted  be- 
tween any  two  given  quantities. 

e.g.  Insert  five  arithmetical  means  between  12  and  3G. 

We  must  find  an  arithmetical  progression  with  five  terms 
between  12  and  36.  Therefore  36  must  be  the  seventh 
term. 

.-.     12  +  6f/  =  36. 

f?  =  4. 

Therefore  the  progression  will  be 

12,  16,  20,  24,  28,  32,  36. 

In  general,  to  insert  n  terms  in  A.  P.  between  a  and  h 
proceed  as  follows: 

Denote  the  common  difference  by  d. 

Then  h,  or  the  (w  +  2)th  term,  is  «  -f  (n  -\-  l)d. 

.'.     rt  + (m  + l)r/  =  ^'. 

.-.     {n^l)d=^b-a. 

1)  —  a 
.  •.     d=  — —r . 
n  -\- 1 

Therefore  the  series  is 

h  —  a        ^  J)  ~  a        ,   J  —  rf  .      I  —  a 

'7^  +  l  M+1  y^-fl  w  +  1 

and  the  required  means  are 

,   h  —  a        .    J)  —  a        .   J)  —  a  .      l  —  a 

a  -1 r^,  a  +  2  -^-^,  a  +  3 — — r, ,  «  +  n — — r, 

^   n-^\  n-\-\  ^  +  1  n-\-r 


or 


na  -]-  b    (n  —  l)a  +  2^     (n  —  2)a  +  3^  a  -{-  nb 

'  ¥+T'  TzT+l         '  n-\-l        ' n-\-  i' 


336  THE  PB0GBE88I0NS. 


EXERCISE  CXXVI. 

1.  Find  the  twentieth  term  of  each  of  the  following 
arithmetical  progressions : 

1°.     7,  10,  13,  etc.  2°.     2,  6,  10,  etc. 

3°.     20,  15,  10,  etc.  4°.     1/12,  1/2,  11/12,  etc. 

2.  Find  the  last  term  of  each  of  the  following  series : 
1°.     4,  7,  10,  to  17  terms.       2°.     3,  7,  11,  to  21  terms. 
3°.     8,  6,  4,  to  12  terms.        4°.     5,  8^,  llf,  to  16  terms. 
5°.     1/3,  -  1/2,  -  4/3,  to  25  terms. 

3.  The  eleventh  term  of  an  A.  P.  is  51  and  the  sixth 
is  31.     What  is  the  first  term  ? 

4.  The  seventh  term  of  an  A.  P.  is  37  and  the  twelfth 
term  is  62.     What  is  the  first  term  ? 

5.  The  fourth  term  of  an  A.  P.  is  10  and  the  tenth 
term  24.     What  is  the  common  difference  ? 

6.  The  sixth  term  of  an  A.  P.  is  5/4  and  the  fifteenth 
term  11/4.     What  is  the  common  difference  ? 

7.  The  third  term  of  an  A.  P.  is  1/2  and  the  thirteenth 
is  2.     What  is  the  twenty-third  term  ? 

8.  The  seventh  term  of  an  A.  P.  is  5  and  the  fifth  term 
is  7.     Wliat  is  the  twelfth  term  ? 

9.  Which  term  of  the  series  6,  11,  16,  etc.,  is  96  ? 

10.  Which  term  of  the  series  7,  3,  —  1,  etc.,  is  —  53  ? 

11.  Which  term  of  the  A. P.   16^^  -  U,  Iba  -  U,  Ua 

-  6b  is  Sa  ? 

12.  Insert  twenty-two  arrithmetical  means  between  8 
and  54. 

13.  Insert  eight  arithmetical  means  between  1  and  0. 


ARITHMETICAL  PliOGBESSION.  337 

14.  Insert  ten  arithmetical  means  between  5a  —  Qb 
and  5^  —  Qa. 

16.  The  sum  of  the  fourth  and  seventh  terms  of  an 
A.  P.  is  40,  and  the  sum  of  the  sixth  and  tenth  is  60. 
Find  the  common  difference  and  the  first  term. 

16.  The  sum  of  the  fifth  and  eleventh  terms  of  an  A.  P. 
is  0,  and  the  sum  of  the  third  and  eighth  terms  is  15. 
Find  the  common  difference  and  the  first  term. 

17.  The  sum  of  the  fourth  and  thirteenth  terms  of  an 
A.  P.  is  —  22,  and  of  the  second  and  eighth  is  24.  What 
is  the  sum  of  the  sixth  and  twelfth  terms  ? 

235.  Problem.  To  find  the  sum  of  any  7iumber  of 
terms  of  a7i  arith7netical  progression. 

Let  a  be  the  first  term,  d  the  common  difference,  n  the 
number  of  terms  whose  sum  is  required,  I  the  last  term, 
and  S  the  required  sum. 

Then,  since  I  is  the  nth  term,  we  have 

I  —  a  -\-  {n  —  l)d. 
,'.     S=a-{-{a-ird)+{a-\-2d)+  .  .  .  -\-{l-2d)-\-{l-d)-{-I, 
or  in  reverse  order, 

S=lJr{l-d)-\-{l-2d)-\-..  .  ^(a+2d)  +  (a+d)-\-a. 
Adding  these  two  equations,  we  obtain 
<^S  —  (a  -\- I)  -\-  {a  -\- 1)  -{-  (a -{- I)  -\- .  . .  to  n  terms 
=  7i(a-\-l ). 

0-.     S=-{a  +  l),  (1) 

But  l  =  a-\-  {n  -  l)d. 

Substitute  this  value  of  I  in  (1),  and  we  get 

8  =  l{U  +  (n-l)d).  (2) 


338  THE  PROGRESSIONS. 

Both  these  formulae  are  important.  By  means  of  the 
second,  when  any  three  of  the  four  quantities  S,  a,  d,  and 
n  are  given,  the  fourth  may  be  computed. 

e.g.  1.  Find  the  sum  of  the  first  thirty  terms  of  the 
series 

3  +  6  +  9,  etc. 

Here  o^  =  3,    d—d^   and  n  =  30. 

.-.     ^=^[6  +  29x  3]  =  1395. 

e.g.  2.  The  sum  of  twelve  terms  of  an  A.  P.  is  260  and 
the  first  term  is  20.    What  is  the  common  difference  ? 

Here  /Sf=260,    n  =  1%,    and   a  =  20. 

.-.     260  =  -^(40  +  116?), 

or  260  =  240  +  QQd. 

.-.     66t^=-20, 

and  d  =  —  -^. 

e.g.  3.  How  many  terms  of  the  series  40  +  36  +  32 
+  etc.  must  be  taken  that  their  sum  may  be  216  ? 

Here  S^UQ,    «  =  40,    and    d  =  -  4r 

.-.     216=|[80  +  (w-l)  X-4], 

or  432  =  80/1  -  4^^2  +  4w. 

.-.  n^-  2l7i  + 108  =  0. 

.-.  {n-  9)(w-12)  =  0 

.\  n  =  9   or   12. 

The  finding  of  the  number  of  terms  by  this  formula  in- 
volves the  solution  of  a  quadratic  equation  in  n,  and  one  or 


ARITHMETICAL  PROOBESSION.  339 

both  of  the  values  of  n  may  be  negative,  fractional,  surd,  or 
imaginary.  In  these  cases  all  the  values  except  the  positive 
integral  ones  must  be  rejected.  When  the  two  values  of  7i 
are  positive  and  integral,  the  sum  of  the  additional  terms 
for  the  greater  value  must  be  zero.  In  the  above  case  the 
tenth,  eleventh,  and  twelfth  terms  are  4,  0,  —  4. 

EXERCISE    CXXVII. 

Find  iiie  sum  of  the  following  series : 

1.  3  +  5  -|-  7  +  . . .  to  twenty-four  terms. 

2.  12  +  llf4-ll|  +  ...to  twenty- two  terms. 

3.  3  +  4i  +  6  +  .  .  .  to  seventeen  terms. 

4.  — 7  —  2-|-3  +  ...to  twenty  terms. 

5.  1/2  +  1/3  +  1/6  +  •  •  •  to  seven  terms. 

6.  5  +  6.2  -[-  7.4  +  .  .  .  to  twenty-one  terms. 

7.  {}i  +  1)  +  {"^n  +  3)  +  (3m  +  5)  +  .  .  .  to  ^i  terms. 

8.  {a  +  hf  +  {a^  J^h^)-\-(a-bf+  .  ..  to  n  terms. 

9.  The  fourth  and  thirteenth  terms  of   an  A.  P.  are 
—  9  and  -{-  9.     What  is  the  sum  of  the  first  twenty  terms  ? 

10.  The  seventh  term  of  an  A.  P.  is  43|  and  the  twelfth 
is  77^.     What  is  the  sum  of  the  first  twenty-four  terms  ? 

11.  Find  the  sum  of  thirty  consecutive  odd  numbers  of 
which  the.  least  is  7. 

12.  Find  the  sum  of  twenty  consecutive  odd  numbers 
of  which  the  greatest  is  77. 

13.  Insert  seventeen  arithmetical  means  between  4  and 
76,  and  find  their  sum. 

14.  Insert  forty  arithmetical  means  between  10  and 
100,  and  find  their  sum. 


340  THE  PROGRESSIONS. 

15.  Find  the  sum  of  all  the  multiples  of  7  lying 
between  200  and  400. 

16.  Find  the  sum  of  all  the  positive  multiples  of  12  of 
less  than  four  digits. 

236.  The  Average  Term. — An  A.  P.  of  an  odd  number 
of  terms  must  contain  a  middle  term,  and  the  number  of 
terms  between  the  first  term  and  this  middle  term  must  be 
the  same  as  that  between  it  and  the  last  term.  Hence  the 
first,  middle,  and  last  terms  must  form  an  A. P.,  and  the 
middle  term  must  be  half  the  sum  of  the  two  extreme 
terms. 

Since  the  formula   S  =  -^{a  -\-  I)   may    be  written 

lit 

S  =  7ii — - — 1,  the  sum  of  an  A.  P.  of  an  odd  number  of 

terms  is  equal  to  the  product  of  the  middle  term  and  the 
number  of  terms.  The  middle  term  therefore  must  be  the 
average  of  all  the  terms,  or  the  arithmetical  mean  of  any 
pair  of  terms  equally  removed  from  it. 

The  average  of  all  the  terms  of  an  A.  P.  evidently  must 
be  half  the  sum  of  the  extreme  terms  or  their  arithmetical 

mean.     For  the  average  of  the  first  and  last  is  ,  of 

2 

the  second  and  next  to  the  last  ^^-^ — ~^  —  ''—^- , 

2  2 

and  so  on. 

Hence,  if  the  number  of  terms  be  odd,  the  average  of 
all  the  terms  will  be  the  middle  term,  and  if  the  number  of 
terms  be  even,  the  average  of  all  the  terms  will  be  the 
arithmetical  mean  of  the  two  middle  terms. 

e.g.  1°.  The  first  term  of  an  A.  P.  of  seventeen  terms 
is  3  and  the  last  term  is  27.    ^Vhat  is  the  sum  of  the  terms  ? 

Here  the  middle  term  :=  —^ —  =  15, 

and  the  sum  =  17  X  15  =  255. 


ARirHMETICAL  PROGRESSION.  341 

e.g.  2°.  The  first  term  of  an  A.  P.  is  17,  the  common 
difference  is  ~  3,  and  the  middle  term  is  —  4.  Find  the 
number  of  terms  and  their  sum. 

Here  -  4  =  17  +  (?i  -  1)  X  -  3, 

-  4  =  17  -  3?i  +  3. 
.  •.     Zn  =  24. 


Since  8  is  the  number  of  the  middle  term,  the  whole 
number  of  terms  must  be  15,  and  their  sum  —  60. 

EXERCISE  CXXVIII. 

1.  Find  the  sum  of  the  twenty-one  terms  of  an  A.  P. 
of  which  the  middle  term  is  33. 

2.  Find  the  sum  of  forty-five  terms  of  an  A.  P.  of 
which  the  twenty- third  is  75. 

3.  The  first  term  of  an  A.  P.  is  3,  the  last  term  is  77, 
and  the  sum  of  the  terms  is  520.  What  is  the  number  of 
terms  ? 

4.  The  first  term  of  an  A.  P.  is  12,  the  last  term  is 
—  198,  and  the  sum  of  the  terms  is  —  3069.  What  is  the 
number  of  the  terms  ? 

5.  A  man  travels  5  miles  the  first  day,  8  miles  the  sec- 
ond, 11  miles  the  third,  and  so  on.  At  the  expiration  of 
a  certain  time  he  finds  he  has  travelled  at  the  average  rate 
of  18^  miles  a  day.     How  many  days  did  he  travel  ? 

6.  A  pedestrian  having  to  go  184  miles  walks  30  miles 
the  first  day,  and  two  miles  less  each  subsequent  day  till 
liis  journey  was  completed.  How  many  days  did  it  take 
him  ? 

7.  In  an  A. P.  .the  product  of  the  sixth  and  eighth 
terms  exceeds  the  product  of  the  fourtli  and  tenth  by  200. 
What  is  the  common  diHerence  ? 


342  THE  PROGRESSIONS. 

8.  In  an  A. P.  the  product  of  the  eighth  and  thirteenth 
terms  is  less  than  the  product  of  the  ninth  and  twelfth 
terms  by  25.     What  is  the  common  difference  ? 

9.  Two  travellers  start  together  on  the  same  road.  One 
of  them  travels  uniformly  at  the  rate  of  10  miles  a  day. 
The  other  goes  8  miles  the  first  day,  and  increases  his  speed 
half  a  mile  each  subsequent  day.  In  how  many  days  will 
the  latter  overtake  the  former  ? 

10.  One  hundred  stones  are  placed  on  the  ground  in  a 
straight  line  at  intervals  of  5  yards.  A  runner  has  to  start 
from  a  basket  5  yards  from  the  first  stone,  pick  up  the 
stones,  and  bring  them  back  to  the  basket  one  by  one. 
How  far  will  he  be  obliged  to  travel  ? 

11.  An  author  wished  to  buy  up  the  whole  edition  of 
1000  copies  of  a  book  which  he  had  published.  He  paid  20 
cents  for  the  first  copy,  but  the  price  rose  so  that  he  was 
obliged  to  pay  1  cent  more  for  each  subsequent  copy  than 
for  the  last.     What  was  he  obliged  to  pay  for  the  whole  ? 

12.  Find  three  numbers  in  A.  P.  the  sum  of  whose 
squares  is  2900,  and  the  square  of  whose  means  exceeds  the 
product  of  the  extremes  by  100. 

13.  Find  four  numbers  in  A.  P.  such  that  the  sum  of 
the  squares  of  the  extremes  equals  464,  and  the  sum  of  the 
squares  of  the  means  equals  400. 

14.  Find  four  numbers  in  A.  P.  such  that  the  product 
of  the  means  shall  exceed  the  product  of  the  extremes  by 
72,  and  the  sum  of  their  squares  shall  equal  280. 

237.  Two  Important  Series  Allied  to  the  Arithmetical 
Series. — Let  Si  denote  the  sum  of  the  first  powers  of  the 
natural  numbers  from  1  to  n,  S^  denote  the  sum  of  their 
squares,  and  S^  the  sum  of  their  cubes.     Then — 


AUITHMETIGAL  PBOGRESSIOK  843 

For  this  is  an  A.  P.  in  which  the  first  term  is  1  and  the 
last  term  is  n  and  the  number  of  terms  is  also  n. 

This  may  be  proved  as  follows : 

{n  +  If  =  n^-^  3^2  _|_  3^  _^  1. 

Writing  1,  2,  3,  etc.,  in  turn  for  7i  in  this  identity,  we 
get 

23  =  13-}- 3.  12  +  3.  1  +  1; 

33  =  23  +  3.22  +  3.2  +  1; 

43  =  33  +  3.  32  +  3.  3  +  1; 

etc.  etc. ; 

[n-\-lY  =  n^^^.n^-\-^.n-\- 1. 

Note  that  we  have  on  each  side  of  these  equations  23,  3^, 
43 ...  to  n^.  Adding  these  equations  and  cancelling  their 
common  terms,  we  get 

0^+1)3=13+3(12+22+32. ..+^^2)_|_3(l_^2+3...  +  w)+?^ 
:.l  +  3>%  +  3^^(^)  +  . 
3^z(MJ_)   ,   2Mi2 

3^^^+^^^+^ 
=  0^2  H ^ . 

)l{n  +  1)3      37^2  +  5^  +  2  2^3  ^  3^2  _^  ^ 

•*•    ^^^^-  2       '^ 2  = 2 

_    ^^^  +  l)(2^  +  l) 
~  2 


S,= 


_     7l{7l  +  1)(2?^  +  1) 

6 


844  THE  PR00IIBS8I0N8. 

3°.  ^3  =  ^i'. 

Here      {n  +  1)*  -  n'^ -\-  4w«  +  6^^  _|_  4^  _^  i. 

Writing  1,  2,  3,  etc.,  in  turn  for  n  in  this  identity,  we 
get 

2^  =  1^  +  4  .  13  +  6  .  12  +  4  .  1  +  1 ; 
34  =  24  +  4  .  23  +  6  .  22  +.4  .  2  +  1; 
44  =  34  +  4  .  33  +  6  .  32  +  4  .  3  +  1; 
etc.  etc. ; 

i^n-\-iy  =  7i^-^^.n^^Q.n^-\-^,n-\- 1. 
Adding  and  cancelling  as  before,  we  get 
{n  +  ly  =  V  +  4.^3  +  6.^%  +  4.>S'i  +  w 

=  1  +  4.  .S3  +  n{n  +  l)(2w  + 1)  +  ^n{n  + 1)  +  w, 
.-.     4.^%=(n+l)4-[^^(r^+l)(27i+l)+247i+l)+^+l] 

=  ^^4  _^  4^^3_^  6^^2  _^  4,^  _^  1  __  [2^3_|_3^2_|_,j^27i2+2/i+ w  +  1] 
=  n^  +  2^i3  _|_  ^2  ^  ,^2(^^  _|.  1)2  3^  [-^(^  _|.  i)-|2, 

...    ^3=[MM^J=^., 

B.    GEOMETRICAL   PROGRESSION". 

238.  Geometrical  Series. — Quantities  are  said  to  be  in 
geometrical  prog7'essio7i  (G.P.)  when  the  ratio  of  any  term 
to  that  which  immediately  precedes  it  is  the  same  through- 
out the  series. 

Thus  each  of  the  following  series  forms  a  geometrical 
progression : 

2,  4,  8,  16,  etc. 

1,  - 1/4,  1/16,  -  1/64,  etc. 

a,  ar,  ar^^  ar^^  etCo 


OEOMETRtCAL  PR00RE88I0N.  S45 

The  constant  ratio  is  called  the  common  ratio,  and  is 
found  by  dividing  any  term  by  the  one  which  immediately 
precedes  it. 

Thus,  in  the  first  of  the  above  series,  3  is  the  common 
ratio;  in  the  second,  —  1/4;  and  in  the  third,  r. 

239.  Type  Form  of  the  Series. — The  type  form  of  a 
geometric  series  is 

a  -\-  ar  -\-  ar^  -{-  ar^  -\-  ar^ . .  .  -{■  af  ~  ^ 

It  will  be  noticed  that  in  this  series  the  exponent  of  r  in 
each  term  is  one  less  than  the  number  of  the  term. 

If  n  denote  the  number  of  terms,  and  I  the  last  or  wth 
term,  then  /  =  ar^  ~  ^ 

240.  Geometrical  Means. — When  three  quantities  are 
in  geometrical  progression,  the  middle  one  is  called  the 
geometrical  mean  between  the  other  two. 

Let  a,  b,  and  c  be  three  quantities  in  G.P.  By 
definition, 

—  =r  — ,   h^  =  aCy    and   h  =  Vac. 
a      h 

That  is,  the  geometrical  mean  between  two  quantities  is 
equal  to  the  square  root  of  their  product. 

All  the  terms  in  a  G.P.  between  the  extremes  may  be 
called  geometrical  means,  and  any  number  of  such  means 
may  be  inserted  between  two  terms. 

Let  a  and  h  be  the  two  terms  between  which  7i 
geometrical  means  are  to  be  inserted. 

The  wliole  number  of  terms  will  be  n  +  3,  and  b  will 
be  the  (w  -f  2)th  term. 

Let  r  be  the  common  ratio.     Then 


ar 


346  THE  PnOGRESSIONS. 

h 


n+  1  _  _ 

a 

n  +  l 


r 


y~a 


e.g.     Insert  four  geometrical  means  between  224  and  7. 
In  this  case  we  must  find  six  terms  in  G.P.  of  which 
the  first  is  224  and  the  sixth  is  7.     Therefore 

7    =224r^ 

.-.     r^  =  l/32, 

and  r  =  y/xj^'l  =  1/2. 

Hence  the  means  are  112,  56,  28,  14. 


EXERCISE    CXXIX. 

In  finding  the  common  ratio  in  a  G.P.  it  is  often 
necessary  to  extract  a  root  of  a  high  index,  which  is  tedious 
without  the  use  of  logarithms.  In  the  following  examples 
it  will  be  easy  to  extract  the  required  roots  by  inspection. 
Kemember  that  the  fourth  root  is  the  square  root  of  the 
square  root,  that  the  sixth  root  is  the  cube  root  of  the  square 
root,  and  that  the  eighth  root  is  the  square  root  of  the 
square  root  of  the  square  root. 

1.  Insert  two  geometrical  means  between  2  and  250. 

2.  Insert  three  geometrical  means  between  —  3  and 
-768. 

3.  Insert  four  geometrical  means  between  5  and— 1215. 

4.  Insert  five  geometrical  means  between  3  and  .000192. 
6.    Insert  four  geometrical  means  between  1/6  and  64/3. 


GEOMETRICAL  PROGRESSION.  347 

241.   Problem.     To  find  the  sum  of  n   terms   of  a 
geometrical  progression. 

Let  8  denote  the  sum,  and  let  the  series  be 


a -\-  ar  -\-  ar  -\-  .  .  .  -\-  ar""  \ 
.  Then        S  =  a  +  ar -{- ar^  ^  .  .  .  +  ar'^'K  (1) 

Multiply  each  side  by  r : 

rS  =  ar  -\-  ar'^  -\-  ar^  -{-...  -\-  ar^.  (2) 

Subtract  (2)  from  (1),  and  we  get 
S  —  rS    —  a  —  ar'^, 
or  (1  -r)S=  a{l  -  r"). 

.-.     S=  a 


1-r 

e.g.  Find  the  sum  of  ten  terms  of  the  series  2  +  4  -f- 
8  +  etc. 

Here  a  =  2,  r  =  2,  and  w  =  10. 

Therefore 

1  oio 

;S'  =  2  A. ^  =  2(210  -  1)  =  2(1023)  =  2046. 

1  —  Z 

242.  Divergent  and  Convergent  Series. — The  formula 

^  /p'"'  ^'* ^  dr^^  ci 

a—, ,    or     a — ,  may  be  written -. 

I  -  r  r  —1         ^  r  —  1      r  —  1 

In  the  series  a  -\-  ar  -\-  ar^  +  «r^  4-  .  .  .  «r"  ~  \.  if  r  be 
made  1,  the  series  becomes  a  +  a  -h  a  -\-  . . .  n  terms  =  7ta. 

Hence,  by  sufficiently  increasing  n,  we  may  cause  S  to 
surpass  any  value  however  great.  When  n  becomes  qo, 
*S'  becomes  oo . 

If  r  be  greater  than  1,  r""  increases  with  n,  and,  by 

sufficiently  increasing  n,   r'^  may  be  made  as  great  as  we 

r"  —  1 
please.     When  w  becomes  oo ,  a— —  becomes  ao  . 


348  THE  PR0ORE88I0N8. 

Hence,  by  sufficiently  increasing  the  value  of  n,  we  may 
cause  S  to  exceed  any  value  however  great,  and  when 
n  =  CO  ,  S  =  cc  . 

In  these  two  cases  the  geometric  series,  if  supposed 
continued  to  an  infinite  number  of  terms,  is  said  to  be 
divergent. 

If  r  be  numerically  less  than  1,  that  is  a  proper  frac- 
tion either  positive  or  negative,  r"  decreases  as  n  increases. 
By  making  n  sufficiently  large  r"^  may  be  made  as  small 

1  —  r" 
as  we   please.       When    n  =  oo ,     r"*  =  0,     and    a-- 

a 


becomes 


1  -  r 
a 


Hence is  the  value  which  8  approaches  as  a  limit 

as  n  is  indefinitely  increased. 

In  this  case  the  series  is  said  to  be  convergent. 

The  sum  of  an  infinite  series  is  the  limit  to  which  the 
sum  of  its  first  n  terms  approaches  as  n  is  indefinitely  in- 
creased. 

If  r  =  —  1,  the  series  becomes 

8=a  —  a-\-a  —  a-\-.,,. 

In  this  case  the  sum  of  any  odd  number  of  terms  is  «, 
and  of  any  even  number  of  terms  0.  The  sum,  therefore, 
does  not  become  infinite  when  an  infinite  number  of  terms 
are  taken,  nor  does  it  converge  to  one  definite  value.  A 
series  which  has  this  property  is  said  to  oscillate,  and  is 
called  an  oscillating  series. 

If  a  series  is  composed  of  an  infinite  number  of  terms, 
its  sum  can  be  found  only  when  the  series  is  converging. 

e.g.     1°.  Find  the  sum  of  the  series 

1/2  +  1/3  +  2/9  +  .  .  .  to  six  terms. 


GEOMETRICAL  PROOttESSION.  349 

1/2(1  -  (2/3)^)  _  1/2(1  -  64/729)  " 
^~         1-2/3         ~  1/3 

_  1/2(665/729)      665 
~  1/3  ~486* 

EXERCISE  CXXX. 

1.  Find  the  sum  of  the  G.P.  6  +  18  +  54  +  .  .  . 
to  eight  terms. 

2.  Find  the  sum  of  the  G.P.  6  -  18  +  54  +  .  .  . 
to  eight  terms. 

3.  Sum  —  2  +  2|  —  3^  +  .  .  .  to  six  terms. 

4.  Sum  3/4  +  li  +  3  +  .  .  .  to  eight  terms. 
6.     Sum  2  —  4  +  8  —  ...  to  ten  terms. 

6.  Sum  16.2  +  5.4  +  1.8  +  .  .  .  to  twelve  terms. 

7.  Sum  —  1/3  +  1/2  —  3/4  +  ...  to  seven  terms. 

8.  Sum  8/5  -  1  +  5/8  -  ...  to  infinity. 

9.  Sum  .45  +  .015  +  .0005  +  ...  to  infinity. 

10.  Sum  1.665  -  1.11  +  .74  -  ...  to  infinity. 

11.  Sum  3-^  +  3-2  +  3-3  -f  .  .  .  to  infinity. 

12.  The  fifth  term  of  a  G.P.  is  324  and  the  eighth  term 
is  -  8748.     What  is  the  first  term  ? 

13.  There  are  five  terms  in  G.  P.  The  sum  of  the  first 
and  second  is  30,  and  the  sum  of  the  fourth  and  fifth  is 
1920.     What  are  the  numbers  ? 

14.  There  are  three  numbers  in  G.  P.  The  sum  of  the 
first  and  second  is  24,  and  of  the  second  and  third  is  —  72. 
What  are  the  numbers  ? 

15.  There  are  three  numbers  in  G.P.  Tlie  second 
minus  the  first  equals  36,  and  the  third  plus  the  second 
equals  210.     What  are  the  numbers  ? 


350  THE  PROGRESSIONS. 

243.  The  Value  of  Repeating  Decimals. — The  value  of 
a  repeating  or  a  recurring  decimal  may  be  found  by  sum- 
ming a  G.P.  to  infinity. 

e.g.  1°.  Find  the  value  of  the  repeating  decimal  .333-|-. 

3         3  3 

•^^^  "^  10  +  10^  +  10^  +  •  •  •  *^ '""^^^^^ 

3  1  3     10      1 


10*  1  -  1/10  "10*  9  ~3' 
Find  the  value  of  the  circulating  decimal  .24L 


_^      41 
~  10  +  102  • 


1         _  W  "I 
..1  -  1/102  ~  99  J 


41       102^  _  ^    .     41 
103  X  99   -  10 

2  X  99  -f  41       239 


10  +  103  X  99   -  10  +  990 


990  990* 

244.  Rule  for  Values  of  Recurring  Decimals.  —  Note 

241 2 

that  the  last  answer  =  — — - — . 
y  yu 

Hence  we  obtain  the  following  arithmetical  rule  for 
finding  the  value  of  a  mixed  circulating  decimal : 

Sithtract  the  non-repeating  figures  from  all  the  digits 
down  to  the  end  of  the  first  period,  and  write  as  a  denomi- 
nator as  many  9'^  as  there  are  digits  i7i  the  repeating  part, 
folloived  hy  as  many  ciphers  as  there  are  digits  in  the  non- 
repeating part. 

Note  also  that  the  answer  to  the  previous  example  = 
3/9.  Hence  we  obtain  the  following  rule  for  finding  the 
value  of  a  pure  recurring  decimal : 


COMPOUND  INTEREST  AND  ANNUITIES.        351 

Write  as  a  doiominator  to  the  recurring  digits  as  many 
9's  as  there  are  digits  in  the  period. 

EXERCISE  CXXXI. 

Sum  the  following  recurring  decimals  as  geometrical 
progressions,  and  show  in  each  case-  that  the  result  is  in 
agreement  with  the  rules  just  given: 


1. 

.15. 

2. 

.185. 

8. 

.396. 

4. 

.428571. 

5. 

.012987. 

6. 

.79. 

7. 

.315. 

8. 

.116.. 

9. 

.19324. 

C.    COMPOUND   INTEREST   AND   ANNUITIES. 

245.  Compound  Interest. — There  are  many  problems  in 
Geometrical  Progression  of  which  an  approximate  solution 
can  be  obtained  readily  by  means  of  logarithms.  Among 
these  the  different  cases  of  compound  interest  and  annuities 
are  of  especial  importance. 

Money  is  said  to  be  invested  at  compound  interest  when 
at  stated  intervals  the  interest  which  has  accrued  is  added 
to  the  principal,  so  as  itself  to  draw  interest.  These  addi- 
tions are  made  usually  annually,  semi-annually,  or  quarterly. 

246.  Problem  I.  To  find  the  amotmt  at  the  end  of  a 
given  time  of  a  sum  of  7noney  invested  at  compound  ititerest 
at  a  given  rate. 

Let  F  denote  the  given  sum, 

71  denote  the  number  of  years, 
r  denote  the  interest  of  one  dollar  for  one  year, 
and        A  denote  the  required  amount. 

1°.  Suppose  the  interest  to  be  computed  annually.  At 
the  end  of  the  first  year  the  amount  will  be 

P  +  riP  =  P(l  +  r); 


352  THE  PROGRESSIONS. 

at  the  end  of  the  second  year  the  amount  will  be 
P(l  +  ^)  -!_  rP(l  +  r)  =  P{\  +  r)(l  +  r)  =  P(l  +  r)^; 

at  the  end  of  the  third  year  the  amount  will  be 
P(l  +  rf  +  rP{l  +  rf  =  P(l  +  r)2(l  +  r)  =  P(l  +  r)^; 

and  at  the  end  of  the  7ith  year  the  amount  will  be 

P(l  +  r)^-'-}-rP(l  +  r)'^-^  =  P(l  +  r)"-i(l  +  r) 

=  P(l  +  r)". 
The  amounts 

P(l  +  r),     P(l  +  rf,     P(l  +  r)^  .  .  .  .  P(l  +  r)% 

are  in  geometrical  progression,  the  first  term  being  P(l+r), 
the  last  term  P(l  +  ry\  and  the  common  ratio  1  +  r. 

^  =  P(l  +  r)^  (1) 

To  solve  this  by  logarithms,  it  is  necessary  to  take  out 
log  P,  log  (1  +  r)%  and  the  antilog  of  the  sum  of  these  two 
logs. 

2°.  If  the  interest  be  computed  semi-annually,  the  for- 
mula for  the  amount  becomes 

A  =  p{l  +  If;  (3) 

and  if  the  interest  be  computed  quarterly,  the  formula  be- 
comes 


A  =  P 


{'+iT'  (^) 


247.  Present  Worth. — The  present  ivortli  of  a  sum  of 
money  due  at  some  future  time  without  interest  is  the 
principal  which  put  at  interest  for  the  given  time  would 
amount  to  the  given  sum. 

248.  Problem  II.  To  find  tlie  present  imrtli,  at  com- 
pound  interest,  of  a  fixed  sum  due  at  a  future  date. 

In  formula  (1),  if  A  denotes  the  given  sum,  r  the  cur- 


COMPOUND  INTEREST  AND  ANNUITIES.        353 

rent  rate  of  interest,  and  n  the  given  number  of  years,  then 
P  will  evidently  denote  the  present  worth.     Hence 

^  =  (]-^»  =  ^(i +  '•)-"•  w 

To  solve  this  by  logarithms,  it  is  necessary  to  take  out 
the  log  of  A,  the  colog  of  (1  +  ^)^  and  the  antilog  of  their 
sum. 

Of  course,  if  the  interest  is  to  be  computed  semi-annu- 
ally or  quarterly,  P  must  be  found  from  formula  (2)  or 
(3). 

249.  Problem  III.  To  find  the  amount  at  a  given 
time  of  a  fixed  sum  invested  at  stated  intervals  at  compound 
interest. 

Let  P  denote  the  fixed  sum,  and  use  A,  r,  and  n  as  be- 
fore. Then  the  amounts  of  the  stated  investments,  on  the 
supposition  that  they  are  made  annually,  will  be  as  follow  : 

A,  =  P(l  +  r)% 

A,  =  P(l  +  rr-\ 

A,  =  P(l^ry-^ 


A^  =  P(l  +  r)"-("-«  =  P(l  +  r). 

The  sum  of  these  amounts  is 

-P(l  +  r)-{-  P(l  +  ry  -h  ^"(1  +  ^)'  + ^(1  +  ry. 

This  is  a  geometrical  progression,  of  which  the  first 
term  is  P(l  +  ^)?  the  common  ratio  (1  +  r),  and  the 
number  of  terms  7i.     Hence 

^  '  1  +  r  —  1  r 

To  solve  this  by  means  of  logarithms,  first  find  by  loga- 


354  THE  PIIOGEESSIONS . 

rithms  the  value  of  (1  +  r)"  +  S  from  this  subtract  1  -|-  r, 
find  the  logarithm  of  the  result,  of  P,  and  the  colog  of  r, 
and,  finally,  the  antilog  of  the  sum  of  the  three. 

250.  Annuities. — An  annuity  is  a  fixed  sum  of  money 
payable  at  equal  intervals  of  time. 

If  the  payment  continue  for  a  definite  time,  the  annuity 
is  called  q.  fixed  annuity  ;  if  only  during  a  person's  life,  a 
life  an7iuity  ;   and  if  for  all  time,  a  perpetuity. 

Annuities  may  pay  annually,  semi-annually,  quarterly, 
or  at  any  other  stated  times,  but  the  principles  of  dealing 
with  all  the  cases  being  the  same,  we  shall  consider  only 
the  case  of  annual  annuities. 

251.  Problem  IV.  To  find  tlie  present  value  of  an 
annuity  of  a  given  amount  payable  at  the  end  of  each  of  n 
successive  years. 

Let  A  denote  the  amount  of  each  payment,  P  the  pres- 
ent worth  of  the  whole  annuity,  and  P, ,  P^,  etc.,  the 
present  worth  of  the  successive  payments,  beginning  with 
the  first.     Then 

P,  =  J(l  +  r)-\ 

P,  =  A{1  +  rY\ 


P^=A{l^rf\ 
mn,^P=A[^^.^^^,-^.....^^J^ 

1 L_ 

In  case  of  a  perpetuity,  n  becomes  oo ,  and  y- — ; — r-  be- 

(1  -f-  ry 

A 
comes  0.     Therefore  P  =  — . 

r 


COMPOUND  INTEREST  AND  ANNUITIES.        355 

That  is,  the  present  ivorth  of  a  perpetuity  is  the  qttotient 
obtained  by  dividinff  the  amount  of  the  anmml  papnent  by 
the  interest  of  one  dollar  for  one  year. 

252.  Problem  V.  To  find  the  amount  of  an  annuity 
to  run  for  n  years  which  can  be  purchased  for  a  given  sum 
of  money,  the  rate  of  compound  interest  being  known. 

In  formula  (6),  P  denotes  the  present  value  or  the  pur- 
chase-money, and  A  the  amount  of  the  annuity.  From 
(6),  we  obtain 


rP  _   rP(l-fr)" 

1- 


(1  -f  rf 

Formula  (7)  is  also  the  formula  for  finding  by  what 
fixed  annual  payment  of  A  dollars  an  obligation  of  P 
dollars  may  be  cancelled  in  a  given  number  of  years, 
r  being  the  interest  of  one  dollar  for  one  year. 

253.  Problem  VI.  To  find  the  present  ivorth  of  an 
annuity  to  begin  after  m  years  and  to  continue  for  n  years, 
allotving  compound  interest. 

By  (6),  the  value  of  the  annuity  at  the  expiration  of  m 
years  is 


(1  + 
and  by  (4),  the  present  worth  of  this  sum  due  in  m  years  is 

A^  _  ^ 

rV        (l  +  rY   _A{(l  +  rY-l) 


356  THE  P1100RES8I0N8. 


EXERCISE  CXXXII. 


1.  What  will  be  the  amount  of  2000  dollars  for  15 
years  at  5  per  cent,  the  interest  being  compounded 
annually  ? 

2.  What  will  be  the  amount  of  800  dollars  for  9  years 
3  months  at  4  per  cent,  the  interest  being  compounded 
quarterly  ? 

3.  What  sum  of  money  will  amount  to  11240.60  in 
5  years  6  months  at  6  per  cent,  the  interest  being  com- 
pounded semi-annually  ? 

4.  In  how  many  years  will  968  dollars  amount  to 
11269.40  at  5  per  cent,  the  interest  being  compounded 
semi-annually  ? 

6.  What  is  the  present  worth  of  a  note  for  600  dollars 
due  9  years  hence,  allowing  4-^  per  cent  compound  in- 
terest ? 

6.  At  what  rate  per  annum  will  2600  dollars  give 
$416.40  in  3  years  and  9  months,  the  interest  being  com- 
pounded quarterly  ? 

7.  In  how  many  years  will  500  dollars  double  itself  at 
5  per  cent,  the  interest  being  compounded  annually  ? 

8.  In  how  many  years  will  a  sum  of  money  double 
itself  at  4  per  cent,  the  interest  being  compounded  quar- 
terly? 

9.  What  is  the  present  value  of  an  annuity  of  500 
dollars  to  continue  for  20  years,  allowing  4  per  cent  com- 
pound interest  ? 

10.  What  is  the  present  value  of  a  perpetuity  of  300 
dollars,  allowing  5  per  cent  compound  interest  ? 

11.  What  is  the  present  value  of  an  annuity  of  400 


HARMONIC  .PROGRESSION.  357 

dollars  to  begin  8  years  hence  and  to  run  for  15  years, 
allowing  4  per  cent  compound  interest  ? 

12.  What  fixed  annual  payment  must  be  made  to  can- 
cel an  obligation  of  3000  dollars  in  8  years,  allowing  3^ 
per  cent  interest  ? 

13.  What  annuity  to  continue  12  years  can  be  pur- 
chased for  4000  dollars,  allowing  5  per  cent  compound 
interest  ? 

D.    HARMONIC    PROGRESSION". 

254.  Harmonic  Progression. — Three  quantities  are  said 
to  be  in  harmonic  progression  when  the  first  is  to  the  third 
as  the  difference  between  the  first  and  second  is  to  the 
difference  between  the  second  and  third.  An  harmonic 
progression  is  denoted  by  the  abbreviation  H.P. 

a,  J),  and  c  are  in  H.P.  when 

a-.l  =  a  —  h:h  —  c. 

A  series  is  said  to  be  harmonic  when  every  three  con- 
secutive terms  are  in  H.P. 

255.  Theorem  I.  If  three  quantities  are  in  har- 
monic progression.)  their  reciprocals  are  in  arithmetical 
progression. 

Let  a,  h,  and  c  be  three  quantities  in  harmonic  pro- 
gression.    Then 

a'.c  ^=  a  —  h'.l)  —  c. 

Whence  «(J  —  c)  =  c{a  —  V), 

or  ah  —  ac  =  ac  —  he. 

Dividing  each  term  by  adc,  we  have 

c      h~  1)      a' 


358  THE  PROGRESSIONS. 

Harmonical  properties  are  interesting  because  of  their 
importance  in  geometry  and  in  the  theory  of  sound.  In 
algebra,  the  theorem  just  proved  is  the  only  one  of  any  im- 
portance. There  is  no  general  formula  for  the  sum  of  any 
number  of  terms  in  H.P.  Questions  in  H.P.  are  solved 
usually  by  taking  the  reciprocals  of  their  tejms,  and  making 
use  of  the  properties  of  the  resulting  A.  P. 

256.  Theorem  II.  The  liarmonic  mean  of  two  quan- 
tities is  equal  to  ticice  their  ])roduct  divided  by  tlieir  sum. 

If  a,  b,  and  c  are  in  H.P.,  — ,    ^,    and    —  are  in  A. P. 

a     b'  c 

.:     ^+1  =  ^. 

a       c      b 

a  -\-  c 

267.  Theorem  III.  The  geometric  mean  of  two  quan- 
tities is  also  the  geometric  mean  of  the  arith7netic  and  har- 
monic mea7is  of  the  quantities. 

Denote  the  arithmetic,  geometric,  and  harmonic  means 
of  a  and  ^  by  J,  0,  and  H,  respectively.     Then 


A  = 

2 

• 

G  = 

Vab. 

H  = 

Ub 

.'.     A 

.  H  = 

ab  = 

0\ 

258. 

Problem. 

To  insert  n 

harmonic  means 

betivem 

a 

and  b. 

Insert  n  arithmetical  means  between  -  and  -=-,  and  the 

a  b 

reciprocals  of  these  will  be  the  required  harmonic  means. 


BARMONIC  PROGRESSION.  359 

EXERCISE  CXXXIII. 

1.  Insert  two  harmonic  means  between  3  and  12. 

2.  Insert  two  liarmonic  means  between  2  and  1/5. 

3.  Find  the  fifth  term  of  the  H.P.  1/2,  1/4,  1/6. 

4.  Insert  three  harmonic  means  between  5  and  25. 

5.  If  a,  h,  c,  are  in  A. P.,  and  i,  c,  d,  are  in  H.P., 
prove  that  a  \h  —  c  -.  d. 

6.  Show  that  if  a,  h,  c,  d,  be  in  H.P.,  then  will 

?>{h  -  a){d  -  c)  =  {c  -  i){d  -  a). 

7.  Show  that  if  .a,  h,  c,  be  in  A. P.,  h,  c,  d,  in  G.P., 
and  c,  d,  e,  in  H.P.,  then  will  a,  c,  e,  be  in  G.P. 


CHAPTER  XXVII. 
BINOMIAL  THEOREM. 
259.  Theorem. —  When  n  is  a  positive  integer, 
(a  +  xY  =  a"  +  na""  -^x-{-  ^^^^a"  "  V 

+  1.2.3         "^      ^ 

-\-  .  .  .  to  n  -\- 1  terms. 
1°.  When  n  =  1,  we  have 
{a -{-  xY  —  a -\-  X  =  a""  -{-  na""  ~  %     since    a^"^  =  «o  =  1. 
By  actual  multiplication,  when  ?^  =  2,  we  have 

{a-\-xy=a^-\-2ax-{-x^=a''+na''  -^x-{-  ^^^^~^V  "  V, 

since  a^'^  —  aP  =  1. 

When  w  =  3,  we  have 

(a-\-xf  =  «3  +  3 A  +  3«a;2  +  a;^  =  «"  +  na""  -^x-\-    \    _    \ 

1 .  z 

When  w  =  4,  we  have 
(a  +  a;)^=  a^  +  4  A  +  Ga^a^^  +  ^ao?  -|_  a;4  =  ^n  _|_  ^^n- 1^ 

y^(^-l)        2  o   ,  n{n-l){n-^) 
+      1.2      "^      ^+  17273         ""      ^* 

^^  -  1)(^  -  2)(^  -  3) 
+  1.2.3.4  "^      ''• 

360 


BINOMIAL  THEOREM.  361 

We  thus  see  that  the  theorem  holds  true  when  n  ^  \, 
2,  3,  or  4. 

2°.     Now  multiply  each  member  of  the  expression 

(a  +  xf  =  fl"  4-  na^'-'^x  +    \  ~  ^U^'-^x^ 

1  .  Z 

+  1.2.3  "^         ^ 

4" .  •  .  to  (w  +  1)  terms, 
which  we  have  found  to  hold  true  when  n  =  l,2,  3,  and  4, 
hy  a  -{-  X,  and  we  obtain 

(a  +  a:)'*  +  i  =  ««  +  !+  [«"a;  +  na^'x] 

-f-  •  •  •  to  7i  +  2  terms. 

Note  that  the  second  term  of  the  last  aggregate  is  ob- 
tained by  multiplying  the  fifth  term  of  the  expression  of 
(a  +  ^)"  hy  a. 

Note  also  that  each  aggregate  contains  two  terms  in  ax 
with  identical  exponents,  and  that,  if  we  let  r  +  1  denote 
the  number  of  the  aggregate,  the  coefficient  of  these  two 
terms  of  each  aggregate  after  the  first  will  be  respectively 

n{n-l),  .  .(n-(r-  1))  n{n  -  1)  .  .  .  (?i  -  r) 

1  .  2  .  .  .  .  r ^""^  1.2.  ...r+1      • 

^(,,  _  1) .  .  .  (^  _  (^  _  1))       n(n-l)..,(n-r) 
1.2.  ...r  "^       1.2.  ...r  +  1 
n{n-l).  .  .(n-(r-  1))     T         n  -  rn 
1.2. ...r  L    ■^r+lj 


362  BINOMIAL   THEOREM. 

^n{n-l)...(7i-{r-l))      n^l 

1.2 r  ^  r  +  1 

_{n  +  l)n{n  -  1)  ...  (w  -  (r  -  1)) 
~  1.  2.  .  .  .  r(r+l)  ' 

whatever  r  may  be,  and  this  is  the  general  expression  for 
the  sum  of  the  coefficients  of  the  term  in  ax  in  each  bracket 
after  the  first. 

Therefore  we  have 

(a  +  xf^^  :=«"  +  !  +  {n  +  l)a''x  +  i^±li)^«~  - 1^^^ 

1  .  Z 

{n^l)n{n-l)     _  (n  ^  l)n(n  -  l)(n  -  ^) 

+  1.2.3         ""      ^  "^  1.2.3.4  "^     ^ 

+  .  .  .  to  («  +  2)  terms. 
If  we  put  n  -\- 1  =  n' ,  WQ  will  have 

1  .  /O 

+  >»'(«' -W-V-%^  +  ...  to  (»'  +  1)  terms, 

which  agrees  with  the  theorem. 

We  therefore  conclude  that  the  theorem  will  be  true  for 
the  next  higher  value  of  n  if  it  be  true  for  any  one  value 
of  n. 

But  by  actual  multiplication  the  theorem  has  been 
shown  to  hold  true  when  ?i  =  1,  2,  3,  and  4.  It  therefore 
niust  hold  true  when  n  =  5,  6,  7,  or  any  positive  integer. 

260.  The  Binomial  Coefficients. — The  quantities 

n{n-J)      n{n  -  l){n  -  2) 
""'         1.2     '  1.2.3         '^^''" 

are  known  as  the  binomial  coefficients. 

Note  that  the  factors  in  the  numerators  begin  with  n 


BINOMIAL  THEOREM.  Z^^ 

and  decrease  by  1,  and  that  their  number  is  one  less  than 
the  number  of  the  term  in  which  it  occurs;  also  that  the 
factors  in  the  denominators  begin  with  one  and  increase  by 
unity,  and  that  the  number  of  factors  in  the  denominator 
is  the  same  as  in  the  numerator. 

e.g.     The  coefficient  of  the  fifth  term  of  the  develop- 

meut  of  (.  +  x)»  is  >K>' -!)(>» -2)(«  -  3)_ 
^  ^  1.2.3.4 

Note  carefully  that  the  binomial  coefficient  of  the  next 
term  in  the  development  of  a  binomial  expression  can  be 
obtained  by  multiplying  the  coefficient  of  the  last  by  the 
exponent  of  a  in  that  term  and  dividing  by  the  number  of 
the  term. 

Thus   the   binomial   coefficient   of  the   third   term   is 

-^ — --^,  and  the  exponent  of  a  is  n  —  2.     The  binomial 
1  .  Z 

coefficient  of  the  fourth  term  is  — ^^ — - — ^^r -.     This  is 

the  coefficient  of  the  third  multiplied  by  (w  —  2)  and  di- 
vided by  3. 

261.  Developments. — AVhen  a  single  algebraic  expression 
is  changed  into  the  sum  of  a  series  of  terms,  it  is  said  to 
be  developed,  and  the  series  is  called  its  development.  A 
development  may  be  true  in  form,  yet  may  equal  the  func- 
tion only  for  certain  special  values  of  x.  No  development 
can  equal  the  function  except  for  the  values  of  x  which 
make  it  convergent. 

EXERCISE  CXXXIV. 

Find  the  binomial  coefficients  of  the  development  of  the 
following  expressions: 

1.    {a^xf.  2.    {a-^xf.  3.    (a-[-xy. 

4.    (a^xf.  5     {a^xY.  6.    {a^x)". 

7.    {a-\-x)\  8.    {a-^xf.  9.    {a  ^  x)\ 


364  BINOMIAL   THEOREM. 

262.  Coefficients. — Note  in  the  above  examples  that 
after  the  middle  of  the  development,  the  coefficients  of  the 
first  half  are  repeated  in  the  reverse  order. 

When  n  is  odd,  the  number  of  terms  in  the  development 
will  be  even.  There  will  be  no  middle  term,  and  the 
coefficients  of  the  terms  each  side  of  the  middle  of  the  series 
will  be  the  same.  When  n  is  even,  the  number  of  terms  in 
the  development  will  be  odd,  and  there  will  be  a  middle 
term  whose  coefficient  will  be  the  largest  of  all. 

263.  Exponents. — Note  also  that  the  sum  of  the  expo- 
nents of  the  two  terms  of  the  binomial  in  each  term  of  the 
development  is  equal  to  7i,  and  that  the  exponent  of  the 
second  term  of  the  binomial  is  always  one  less  than 
the  number  of  the  term  in  which  it  occurs  in  the  develop- 
ment. The  exponent  of  the  first  term  will  be  n,  minus  the 
exponent  of  the  second  term. 

e.g.  In  the  sixth  term  of  the  development  of  (a  +  xy 
we  have  a'^x^. 

264.  Signs. — When  both  terms  of  the  binomial  to  be 
developed  are  positive,  all  the  terms  of  the  development  are 
positive,  since  all  powers  of  positive  quantities  are  positive. 

When  the  first  term  of  the  binomial  is  positive  and  the 
second  term  negative,  every  other  term  of  the  development 
beginning  with  the  second  is  negative. 

e.g.  Write  the  product  of  the  powers  of  the  first  and 
second  terms  of  the  binomial  (c  —  2x^y  in  the  fourth  term 
of  its  development. 

c%-  2xY  =c^X  -Sx'=  -  8(^x\ 

EXERCISE   CXXXV. 

Write  the  product  of  the  powers  of  the  two  terms  of  the 
following  binomials  in  the  given  term  of  their  develop- 
ment. 

N.B. — When  the  terms  of  the  binomial  expression  to  be 
developed  are  complex,  they  should  in  all  cases  be  thrown 


BINOMIAL   THEOREM.  365 

with  their  signs  within  parentheses,  the  powers  to  which 
these  are  to  be  raised  should  be  indicated,  and  the  binomial 
co'efficient  should  be  written  before  and  then  the  indicated 
operation  should  be  performed. 

1.  In  the  fifth  term  of  (a  +  2x^y\ 

2.  In  the  fourteenth  term  of  (3  —  aY^. 

3.  In  the  fourth  term  of  (5^^  —  Ix^)"*. 

4.  In  the  eighth  term  of  (6a  —  x/by^. 

I  1  V^ 

5.  In  the  seventh  term  of  (2a;  —  —  1   . 

/  1    Y' 

6.  In  the  eleventh  term  of  \\.x . 

\  2  ^/xl 

265.  Practical  Rules.  —  The  work  of  developing  a 
power  of  a  binomial  is  facilitated  by  the  following  arrange- 
ment: 

1°.  In  one  line  write  all  the  powers  of  the  first  term 
beginning  with  the  ^^th  and  ending  with  the  0th,  or  unity. 

2°.  Under  these  write  the  corresponding  powers  of  the 
second  term,  beginning  with  the  0th,  or  unity,  and  ending 
with  the  wth. 

3°.  Under  these,  in  a  third  line,  write  the  binomial 
coefficients. 

4°.  Form  the  continued  product  of  each  column  of 
three  factors,  and  connect  these  products  with  the  proper 
signs.     The  result  will  be  the  required  development. 

e.g.     Develop  (2«  —  Zx^y. 

Powers  of  2a,     32as  +  16a4     +8a3        +  4«5         -f2«        +1. 
Powers  of  -  Soj^    1  -  S**       +  Oa;*        —  27aj8       _|-  81*8     _  243«io. 
Binom.  Coef.  1 -f- 5  +10  +10  +5  +1. 

(2a  -  Zx'f  =     S2a'  -  UOa*x'-\-  T20a^x*~  1080aV+  SlOax^-  24'dx^^ 

Perhaps  the  easiest  way  to  write  out  a  binomial  expres- 


366 


BINOMIAL   THEOREM. 


sion  is  first  to  throw  tlie  complex  terms  with  their  signs 
within  parentheses,  indicate  the  powers  to  which  these  are 
to  be  raised,  and  then  find  the  binomial  coefiicients  by 
successive  applications  of  the  rule  already  given  for  finding 
the  coefficient  of  the  next  term  to  the  one  already  ob- 
tained. 

EXERCISE   CXXXVI. 

Develop  the  following  expressions : 


1. 

{a  +  x)\             2. 

{a 

-xy.        3.  {i-^xy. 

4. 

(^-3)^        5. 

(3. 

:+2#.          6.    {"^x-yy. 

7. 

(1  -  da'Y.         8. 

(1 

-xyy.          9.    (3«-2/3)«. 

10. 

p      3y 
V  3  ^  2^-y   . 

11.       (c^/3  +  f^  3/4)4 

12. 

(m-  V2  _  n^y. 

13.    {x^^  -  2y^"y. 

14. 

[a^  +  5  Vxf. 

15.     (  Va^  +  -i  ^ay. 

16. 

(xy^-^dy-'^r-y. 

17.       («V2^- 2/3  4-^,-1/2^2/3)7^ 

18. 

(1  -  l/xr. 

-  266.  The  General  Term. — The  general  term  of  the 
development  of  {a  -\-  xy  is  usually  designated  the  rth 
term,  r  standing  for  the  number  of  the  term. 

In  any  term  of  the  development  of  {a  +  x)'^ : 

1°.  The  exponent  of  x  is  one  less  than  the  number  of 
the  term. 

2°.  The  exponent  of  a  is  n  minus  the  exponent  of  x. 

3°.  The  last  factor  of  the  numerator  is  one  greater  than 
the  exponent  of  a. 

4°.  The  last  factor  of  the  denominator  is  the  same  as 
the  exponent  of  x. 

Therefore,  in  the  rth  term, 

The  exponent  of  x  will  be  r  —  1 ; 


BINOMIAL   THEOREM.  367 

The  exponent  of  a  will  be  y^  —  (r  —  1)  or  7^  —  r  +  1 ; 
The  last  factor  of  the  numerator  will  be  ';^  —  r  -}-  ^5 
The  last  factor  of  the  denominator  will  be  r  —  1. 
Hence  the  formula  for  the  rth  term  is 

n{n  -  \){ii  -  2)  .  .  .  (^  -  r  +  2) 

1.2.3      r:":    V^^W  ' 

e.g.     The  seventh  term  of  (2aV2  _  I-  2)12. 
In   this   case  n  —  12  and  r  =  7;    hence  the  seventh 
term  will  be 

12  .  11  .  10  .  9  .  8  .  7 


=  924 .  (64«3Z'-i2)  ^  59136«3Z>-i2. 

EXERCISE  CXXXVII. 

1.  Find  the  fourth  term  of  {x  —  5)^^. 

2.  Find  the  tenth  term  of  (1  +  1x)^. 

3.  Find  the  twelfth  term  of  (2a;  -  \y^. 

4.  Find  the  fourth  term  of  («/3)  +  Uy^. 

5.  Find  the  fifth  term  of  (2«  -  1/^)^. 

6.  Find  the  seventh  term  of  {— —1  . 


-2\6 


7.    Find  the  fifth  term 


/^3/2  y5/2\8 


8.-  Find  the  value  of  (x  +  V2y  -\-  {x  -  V2Y. 

9.  Find  the  value  oi  (V2 -]-  ly  -  {  V2  -  If. 

10  Find  the  value  of  [2  -  ^{1  -x)f+[2-\-  ^(l-x)Y, 

11.  Find  the  middle  term  of  {a/x  -f-  x/aY^. 

12.  Find  the  two  middle  terms  of  ida ^ j  . 

(o  1    \  9 

—  X^   —   —- 
^  dx 


368  BINOMIAL   THEOREM. 

267.  Binomial  Theorem  for  any  Rational  Index. — We 

have  seen  that  when  n  is  a  positive  integer,  the  binomial 
function  develops  into  a  finite  series,  the  number  of  whose 
terms  is  w  -f  1.  This  is  because  the  factor  n  —  r  -\-  1 
vanishes  when  r  —  n  -\-  1. 

Now  as  r  is  necessarily  integral,  n  —  r  -{-  1  cannot 
vanish  for  any  fractional  or  negative  value  of  n.  Hence 
when  n  is  negative  or  fractional,  a  function  when  devel- 
oped by  the  binomial  theorem  must  produce  an  infinite 
series  of  terms. 

It  is  shown  in  Higher  Algebra  that  the  development  is 
true  in  form  for  all  rational  values  of  n.  It  must,  however, 
be  borne  in  mind  that  the  series  is  in  reality  an  expansion 
of  the  function  only  for  those  values  of  x  which  render  the 
series  convergent. 

EXERCISE  CXXXVIII. 

Develop  each  of  the  following  binomials  to  five  terms: 
1.    (a  -  x)y\      2.    {a  +  x)y\       3.    (1  -  xY\ 


4.  (1  -f  xy^     5.  (3  -  ^x)y\   6.  1/  n  -  x. 

7.    1/  v'lT^.     8.    y{x^  +  %).     9.    («'  -  ^x-  V2)  -  y\ 


CHAPTER  XXVIII. 
PERMUTATIONS  AND  COMBINATIONS. 

268.  Permutation. — To  permute  a  group  of  things  is  to 
arrange  them  in  a  different  order,  and  the  various  different 
orders  in  which  the  things  in  a  group  may  be  arranged  are 
called  the  im'mutations  of  the  group. 

Thus  I  permute  the  group  formed  by  the  three  letters 
abc  when  I  change  their  order  into  acb,  and  the  six  differ- 
ent orders  in  which  the  letters  of  this  group  may  be  written 
are  called  the  permutations  of  this  group.  These  permuta- 
tions are 

abc,    ach,    hca,    hac,    cab,    cba. 

269.  Combination.  —  To  combine  a  given  number  of 
things  into  groups  each  of  which  shall  contain  the  same 
number  of  things  is  to  select  from  the  whole  the  requisite 
number  of  things  and  put  them  together  without  regard  to 
the  order  in  which  they  are  placed,  and  the  various  groups 
that  may  be  formed  in  this  way  out  of  the  whole  number 
are  called  the  combwations  of  the  things. 

Thus  the  four  letters  a,  b,  c,  d,  may  be  combined  two  at 
a  time,  or  by  twos,  in  six  different  ways,  namely, 

ab,     ac,     ad,     be,     bd,     cd. 

If  the  letters  were  taken  three  at  a  time,  or  by  threes, 
it  would  be  possible  to  make  only  four  combinations, 
namely, 

abc,     aM,     acd,     bed. 


370  PERMUTATIONS  AND   COMBINATIONS. 

270.  Symbols  of  Combination  and  Permutation.  —  If 

the  whole  number  of  things  at  our  disposal  be  denoted  by 
w,  and  the  number  to  be  put  into  each  group  be  denoted 
by  r,  then  the  number  of  possible  combinations  will  be  de- 
noted by  the  symbol  ^C^.  This  symbol  is  read,  oi  things 
combined  by  r's. 

Thus  in  the  above  example 

and  'C,  =  4. 

When  things  are  combined  by  2's  there  are  two  possible 
permutations  for  each  group.  Thus  we  may  write  ab,  or 
ba. 

Of  the  four  letters  a,  b,  c,  d,  the  possible  combinations 

by  2's  are 

ab,   ac,  ad,  be,  bd,  cd. 

Of  each  of  these  groups  there  are  two  possible  permu- 
tations. Hence  the  possible  permutations  of  the  four  letters 
by  2's  are 

ab,   ac,  ad,  be,  bd,  cd, 

ba,  ca,  da,  cb,  db,  dc  —  12. 

Of  the  same  four  letters  the  possible  combinations  by 

3's  are 

abc,     abd,     acd,     bed. 

Of  each  of  these  groups  there  are  six  possible  permuta- 
tions. Hence  the  possible  permutations  of  the  four  letters 
by  3's  are 


abc, 

abd, 

acd. 

bed. 

acb. 

adb. 

adc, 

bde. 

bea, 

bda, 

cda. 

cdb, 

bac, 

bad, 

cad. 

cbd. 

cab. 

dab. 

dac, 

dbc, 

cba, 

dha^ 

dca. 

deb  ^  24. 

PERMUTATIONS  AND   COMBINATIONS.  37l 

In  any  case,  the  number  of  permutations  is  equal  to 
the  product  of  the  number  of  combinations  and  the  number 
of  permutations  of  each  combination. 

Using  n  and  r  as  above,  the  number  of  permutations 
that  are  possible  is  denoted  by  the  symbol  "P^. 

Thus,  ^P,  ^  12     and     ^P,  =  24. 

271.  Number  of  Permutations. — The  important  fact  to 
which  attention  was  called  a  short  time  since  may  be  sym- 
bolized thus:   - 

«P^  =r  "6;  X  ^P,. 

This  is  a  special  case  of  the  following  general  principle : 
If  one  operation  can  be  j)erformed  in  m  ways,  and  if  after 
it  has  been  performed  in  any  one  of  these  ways  a  second 
operation  can  be  performed  in  n  ways,  the  number  of  ways 
of  performing  the  two  operations  will  be  m  X  n. 

The  truth  of  this  statement  is  evident.  For  there  will 
be  n  ways  of  performing  the  second  operation  for  each  way 
of  performing  the  first;  that  is,  n  ways  of  performing  the 
two  for  each  way  of  performing  the  first ;  and  as  there  are 
m  ways  of  performing  the  first,  there  must  be  m  X  n  ways 
of  performing  the  two. 

e.g.  There  are  ten  steamers  plying  between  Liverpool 
and  Dublin.  In  how  many  ways  can  a  man  go  from  Liver- 
pool to  Dublin  and  return  by  a  different  steamer  ? 

There  are  ten  ways  of  making  the  first  passage,  and 
with  each  of  these  is  a  choice  of  nine  ways  of  returning. 
Hence  the  number  of  possible  ways  of  making  the  two 
journeys  is  10  X  9  ==  90. 

This  principle  applies  also  to  the  case  in  which  there 
are  more  than  two  operations  each  of  which  may  be  per- 
formed in  a  given  number  of  ways. 

e.g.  Three  travellers  arrive  at  a  town  in  which  there 
are  four  hotels.  In  how  many  ways  can  they  find  accommo- 
dation, each  at  a  different  hotel  ? 


372  PERMUTATIONS  AND   COMBINATIONS. 

The  first  tniveller  has  a  choice  of  four  hotels,  and  after 
he  has  made  his  selection  in  any  one  way,  the  second  has  a 
choice  of  three.  Hence  the  first  two  can  make  their  choice 
in  4  X  3  ==  12  ways.  With  any  one  of  these  selections,  the 
third  can  select  his  hotel  in  two  ways.  Hence  the  possible 
number  of  ways  is  4  X  3  X  2  =  24. 

272.  Peoblem  I.  To  find  the  number  of  permutations 
of  n  dissimilar  things  taken  r  at  a  time. 

This  is  equivalent  to  finding  in  how  many  different 
ways  we  may  put  one  thing  in  each  of  r  places  when  we 
have  n  different  things  at  our  disposal. 

Evidently  we  may  select  any  one  of  the  n  objects  for 
the  first  place;  hence  we  may  fill  that  place  in  n  different 
ways.  After  any  object  has  been  selected  for  the  first  place 
there  remain  n  —  1  objects,  any  one  of  which  may  be  se- 
lected for  the  second  place.  Hence  the  first  two  places  may  • 
be  filled  in  7i{n  —  1)  different  ways.  After  any  selection 
has  been  made  for  the  first  two  places  there  remain  n  —  2 
objects,  any  one  of  which  may  be  selected  for  the  third 
place.  Hence  the  first  three  places  can  be  filled  in 
n{n  —  l)(vi  —  2)  different  ways.     And  so  on. 

Notice  that  a  new  factor  is  introduced  for  each  place 
that  is  filled,  so  that  the  number  of  factors  will  be  equal 
always  to  the  number  of  places  filled. 

Notice  also  that  the  first  factor  is  the  number  of  objects 
at  our  disposal,  and  that  each  subsequent  factor  is  dimin- 
ished by  unity,  so  that  each  factor  is  the  number  of  things 
at  our  disposal  diminished  by  a  number  which  is  one  less 
than  that  of  the  corresponding  place.  Hence  the  rth  fac- 
tor will  hQ  71  —  {r  —  1)  =  71  —  r  -\- 1. 

Hence  the  number  of  permutations  of  w  things  taken  r 
at  a  time,  or  "P^  =  7i{7i  —  l){^n  —  2)  ...  r  factors, 

or  "P^  =  7l{7l  -  1){71  -  2)  ...  (71  -  7-  +  1). 

When  r  in  the  above  formula  for  the  number  of  per- 


PERMUTATIONS  AND   COMBINATIONS.  373 

mutations  equals  n^  the  last  factor  becomes  1,  and  the  for- 
mula becomes 

-P,^  =  n{n  -  l)(?^  -  2)  ...  3  .  2  .  1. 
This  product  is  (idXlQdi  factorial  n.  It  is  usually  denoted 
by  the  symbol  \n,  or    n\ 

e.g.     1°.   Six  persons  enter  a  room  in  which  there  are 
six  chairs.     In  how  many  ways  may  they  be  seated  ? 
Here  we  have 

«Pe=|G  =  6X5X4X3X2X1  =  720. 

e.g.     2"".  Five  persons  enter  a  room  where  there  are 
eight  chairs.     In  how  many  ways  may  they  be  seated  ? 
Here  we  have 

sp,  =  8X7X6X5X4  =  6720. 

e.g.     3°.  How  many  different  numbers  of  six   digits 
may  be  formed  out  of  the  nine  digits  1,  2,  3,  ...  9  ? 
Here  we  have 

9Pg  =  9X8X7X6X5X4  =  60480. 

273.  Problem  II.  To  find  lioiv  many  of  the  permuta- 
tio?is  ^Pr  contain  a  particular  ohject. 

Denote  the  objects  by  the  letters  of  the  alphabet. 

Find  first  how  many  permutations  there  are  of  all  the 
letters  b4it  a  when  taken  r  —  1  at  a  time.  Then  associate 
a  with  each  of  these  in  every  possible  way.  The  result  of 
these  two  operations  must  be  all  the  permutations  of  the  n 
letters  taken  r  at  a  time  which  contain  the  letter  a. 

The  permutations  oi  n  —  1  things  taken  r  —  1  at  a 
time  are 

"-ip^_i  =  {n  -  l){n  -  2)  .  .  .  {n  -  r  -\-  1). 

In  each  of  these  groups  a  can  have  r  positions,  since  it 
may  occur  first,  or  last,  or  in  every  intermediate  position 
between  the  letters  of  each  group. 


874  PERMUTATIONS  AND   COMBINATIONS. 

Hence  the  number  of  permutations  which  contain  the 
letter  a  is 

r{n  -  l){n  -  2)  .  .  .  (w  -  r  +  1). 

In  a  similar  way  we  may  find  that  the  number  of  per- 
mutations which  contain  two  objects  or  letters  is 

r{r  -  l){n-  2)  .  .  .  {n  -  r  +  1). 

For  if  the  two  letters  a  and  b  be  left  out  and  the  re- 
maining letters  are  arranged  in  groups  of  r  —  2  letters,  the 
number  of  permutations  would  be 

(n-  2){n-d).  ,  .  (n  -  r  +  1). 

Since  each  of  these  groups  contains  r  —  2  letters,  i  may 
be  associated  with  each  in  r  —  1  different  ways.  Hence 
the  number  of  permutations  which  contain  b  would  be 

(r  -  l){7i  -  2){?i  -  3)  .  .  .  {71  -  r  4-  1). 

As  each  of  these  groups  contains  r  —  1  letters,  a  may 
be  associated  with  it  in  r  different  ways.  Hence  the  num- 
ber of  permutations  which  contain  a  and  b  would  be 

r{r  -  l){n  -  2){7i  -  3)  .  .  .  (w  -  r  +  1). 

In  a  similar  way,  the  number  of  permutations  contain- 
ing three  objects  or  letters  would  be 

r{r  -  !)(/•  -  2)(7i  -  S)  .  .  .  (n  —  r  -{-  1), 

etc.  etc, 

e.g.  How  many  numbers  of  four  digits  can  be  formed 
out  of  the  six  digits  1,  2,  3,  4,  5,  6  ?  How  many  of  these 
will  contain  1  ?  How  many  will  contain  1  and  2  ?  How 
many  will  contain  1,  2,  and  3  ? 

1°.  ep,  =  C)  X  5  X  4  X  3  =  360. 

2°.  r{7i  -  l){7i  -  2)(7i  -3)  =  4X5X4X3  =  240. 

3°.  r{r  -  l)(7i  -  2){7i  -3)  =  4x3x4x3  =  144. 

4°.  r{r  -  l)(r  -  2)(w  -  3)  =  4  x  3  X  2  X  3  =  72. 


PERMUTATIONS  AND   COMBINATiONS.  375 

274.  Peoblem.  To  find  the  number  of  permutations 
of  n  tilings  all  together,  ivhen  u  of  the  things  are  alike. 

Denote  the  required  number  of  permutations  by  x. 
Now  if  the  u  things  were  all  unlike  they  would  give  rise  to 
"Pu,  or  u\,  permutations,  each  one  of  which  might  be  com- 
bined with  the  X  permutations,  and  thus  give  rise  to  "P„ , 
or  n !,  permutations.     Hence 

^up     —  np 

n\ 
or  X—  —,-. 

ui 

Similarly,  if  among  the  n  objects  there  were  u  alike  of 
one  kind  and  v  of  another,  then 

^      up        vp     —  np 

•^  '     -L  u  '     -^   V  —     -'■   nf 

n\        , 
or  X  =^  —. — r,  etc. 

u\  v\ 

e.g.  How  many  permutations  can  be  made  from  the 
letters  in  the  word  Mississippi  ? 

Here  there  are  11  letters  in  all,  and  among  them  4  s's, 
4  i's,  and  2ys. 

_       11!  11.10.9.8.7.6.5.4.3.2.1 

^  ~  4!  4!  2l  ~       4.3.2.1.4.3.2.1.2.1 
=  34650. 

If  the  permutations  were  to  contain  no  repeated  letters, 
the  number  of  different  letters  being  4,  the  permutations 
would  be 

*P,  =  4 .  3  .  2  .  1  --=  24. 

EXERCISE  CXXXIX. 

Find  the  value  of : 
1.      '^Pr  .       2.     i^Pg.  3.     'Pr 

4.     How  many  permutations  can  be  made  of  the  letters 
in  the  word  number  ? 


376  PERMUTATIONS  AND  COMBINATIONS. 

6.     How  many  permutations  can  be  made  of  the  letters 
in  the  word  q^iadruple  ? 

6.  How  many  permutations  can  be  made  of  the  letters 
in  the  word  priiiciple  ? 

7.  In  how  many  ways  may  4  red,  3  blue,  and  5  white 
cubes  be  arranged  in  a  pile  ? 

8.  In  how  many  ways  can  7  cards  each  of  a  different 
prismatic  color  be  arranged  in  piles  of  4  cards  each  ? 

9.  How  many  of  these  piles  would  contain  red  ? 

10.  How  many  of  them  would  contain  red  and  green  ? 

11.  How  many  of  them  would  contain  red,  green,  and 
blue? 

12.  A  pack  consists  of  8  white,   6  red,  and  4  blue 
cards.     In  how  many  ways  may  they  be  arranged  ? 

275.  Problem.     To  find  the  number  of  combinations 
of  n  things  tahen  r  at  a  time. 
As  we  have  already  seen, 

nr  -  !^  -  ^(^^  -  l){n  -  2)  .  .  .  (^  -  r  +  1) 

\l~  '  t 

e.g.     How  many  different  committees  of  8  persons  each 
can  be  formed  out  of  a  board  of  16  men  ? 

Here    .^^^16.15.14.13.12.11.10.9 


8.7.6.5.4.3.2.1 

=  12870. 


276.  Problem.  To  find  the  number  of  ti^nes  any 
'particular  object,  a^  will  be  present  in  **  6^. 

If  we  form  "~  ^6V_i  combinations  from  all  the  objects 
except  a  taken  r  —  1  togetlier  we  can  place  a  with  each  of 
these  groups,  and  thus  form  all  the  combinations  of  the 


PERMUTATIONS  AND   COMBINATIONS.  377 

n  objects  taken  r  together  which  contain  a.  Hence  a 
occurs  in  '*"^6'^_,  of  the  combinations.  Similarly,  two 
particular  objects  will  occur  in  ""^(7^_2  of  the  combina- 
tions; etc. 

e.g.  Out  of  a  guard  of  14  men,  how  many  different 
squads  of  6  men  can  be  drafted  for  duty  each  night  ? 

In  how  many  of  these  squads  would  any  one  particular 
man  be  ? 

In  how  many  of  these  squads  would  any  two  given 
men  be  ? 

1°.   ^^C'e  =  3003. 

2°.  '^C,  =  1287. 

3°.  126;  =  495. 

e.g.  From  10  books  in  how  many  ways  can  a  selection 
of  4  books  be  made,  1°  when  a  specified  book  is  included, 
2°  when  a  specified  book  is  excluded  ? 

1°.  Since  one  book  is  to  be  included  in  each  selection, 
we  have  only  to  choose  3  out  of  the  remaining  9. 

'C,  =  84. 

2°.  Since  one  book  is  always  to  be  excluded,  we  must 
select  the  4  books  out  of  the  remaining  9. 

»C;  =  126. 

EXERCISE  CXL. 

1.  In  a  certain  district  4  representatives  are  to  be 
elected,  and  there  are  8  candidates.  In  how  many  differ- 
ent ways  may  a  ticket  be  made  up,  each  ticket  to  contain 
four  names  ? 

2.  Out  of  9  red  balls,  4  white  balls,  and  6  black  balls, 
how  many  different,  combinations  may  be  formed  each  con- 
sisting of  5  red  balls,  1  white  ball,  and  3  black  balls  ? 

Out  of  the  9  red  balls  126  combinations  may  be  formed 


378       '  PERMUTATIONS  AND  COMBINATIONS. 

each  containing  5  balls.  Each  of  these  may  contain  one  of 
the  4  white  balls,  and  there  may  be  formed  20  combinations 
out  of  6  black  balls  taken  2  at  a  time.  As  each  of  these 
may  be  combined  with  the  126  previous  groups,  hence  the 
combinations  will  equal 

126  X  4  X  20  =  10080. 

3.  How  many  combinations  can  be  formed  out  of  5  red, 
7  white,  and  6  blue  objects,  each  combination  to  consist  of 

3  red,  4  white,  and  2  blue  objects  ? 

4.  On  the  supposition  that  the  colored  objects  of  each 
set  are  all  of  different  shape,  how  many  permutations  of 
these  objects  could  be  formed  with  3  red,  4  white,  and  2 
blue  in  each  resulting  set  ? 

6.  Out  of  12  doctors,  15  teachers,  and  10  lawyers,  how 
many  different  committees  can  be  formed,  each  containing 

4  doctors,  5  teachers,  and  3  lawyers  ? 

6.  There  are  fifteen  points  in  a  plane  no  three  of  which 
are  in  a  line.  How  many  ^triangles  can  be  formed  by  join- 
ing them  in  threes  ? 

277.  Meaning  of  the  Binomial  Coefficients. — 

{a  +  xy  =  [a  -\-  x)(a  -\-  x)  =  a^ -\-  2ax  +  x^; 
{a  +  xY  ={a-]-x){a  +  x){a  -^x)  =  a^-{-  da^x+dax^  +  x^; 
{a  -j-  xy  z=z  (a  -\-  x){a  +  ^)(«  +  ^)(«  +  x) 
=  «^  4-  4  A  +6ftV  _^  4^^3  _^  ^4. 

(a  -\-  xy  =  (a  -\-  x){a  -\~  x)  .  .  .  n  factors 
=  a^+  na^-'x  +  ^^^^^~  "^V  "  V .  .  .  to  w  +  1  terms. 

These  products  are  formed  by  taking  a  letter  from  each 

of  the  7i  factors  and  combining  them  in  every  possible  way. 

We  may  take  an  a  from  each  and  combine  these  n  a's 


PERMUTATIONS  AND  COMBINATIONS.  379 

into  a  product  in  every  possible  way.  As  the  letters  are 
all  alike,  there  is  only  one  way  of  combining  them.  Hence 
«"  is  one  term  of  the  product. 

The  letter  x  can  be  taken  once,  and  a  the  remaining 
{n  —  1)  times,  and  the  number  of  combinations  of  «"~^ 
and  X  will  be  the  number  of  ways  in  which  x  may  be  taken 
out  of  the  n  factors,  and  this  is  the  number  of  ways  of 
taking  n  things  1  at  a  time,  or  ^C\  =  n.  Hence  the  term 
«"  "  ^x  will  occur  "Ci  times  and  we  have 

Again,  the  letter  x  can  be  taken  twice,  and  a  the  re- 
maining {n  —  2)  times,  and  the  number  of  ways  in  which 
2  x's  can  be  taken  is  the  number  of  ways  of  taking  n  things 

2  at  a  time,  or  "62  =  — — — ^ — - .     Hence  the  term  «"  ~  "^x^ 

will  occur  ^6^2  times,  and  we  have 

"6'2««-V. 

And,  in  general,  x  can  be  taken  r  times  (r  being  a 
positive  integer  not  greater  than  n),  and  a  the  remaining 
{n  —  r)  times,  and  the  number  of  ways  in  which  r  x'%  can 
be  taken  is  the  number  of  ways  of  taking  n  things  r  at  a 
time,  or 

_  n{n  -  l){n  -  2)  .  .  .  (n  -  (r  -  1)) 
^^  -  1  .  2  .  3  .  .  . .  r 

Hence  we  shall  have     ^C^a  ™  ~  ^x^. 

Hence    («  +  a:)"  =  «"  -f  ^  C^a''  "  ^a;  +  "  C^a''  -  ^x^ 

+  .  .  .  wCVa"- ^a;^  +  ...  to  [~Cna"-"2:^  =  x""]. 

We  thus  see  that  the  binomial  coefficients  are  simply 
•the  number  of  different  ways  in  which  71  things  can  be 
taken  1,  2,  3,  .  .  .  up  to  w  at  a  time. 

They  are     1,  ~(7i,  ^C^,  "C'a,  ...  ~6;  ...  up  to  "C„. 


380  PERMUTATIONS  AND  COMBINATIONS. 

They  are  often  written  C^,  Ci,  C\,  C^, .  .  .Cr,..Cnj  Cq 
being  understood  to  be  1. 

If  we  make  both  a  and  x  equal  to  1,  the  formula 
becomes 

(1  + 1)-  =  1  +  6;  +  c;  +  c; . . .  +  a . . .  +  c'^, 

or       2*^  rr  1  +  6\  +  ^2  +  Cg .  .  .  +  C;  .  .  .  +  Chi. 

That  is,  the  sum  of  the  binomial  coefficients  in  any 
expression  io  n  -{-  1  terms  is  equal  to  2^  —  1. 

Or  the  sum  of  all  the  possible  ways  of  taking  n  things 
1,  2,  3,  up  to  n  at  a  time  is  equal  to  2"  —  1. 


CHAPTER  XXIX. 
DEPRESSION  OP  EQUATIONS. 

278.  General  Equation  of  wth  Degree  in  x. — The  most 
general  form  of  an  integral  equation  of  the  nth.  degree  in 

X  is 

in  which  n  is  a  positive  integer. 

If  we  divide  this   equation  through  by  Jo  7   and   put 

—^  =z  ai,  -J  —  a^,  etc.,  we  obtain 

xn  +  a.x''-''  +  «,x"-2  +  .  .  .  an._,x  +  «^  =  0,  (1) 

which  we  will  consider  as  the  general  form  of  an  integral 
equation  of  the  nth  degree  in  x. 

The  coefficients  «i,  a.^,  etc.,  may  be  integral,  fractional, 
or  surd,  but  we  shall  consider  only  the  cases  in  which  these 
coefficients  are  rational. 

If  none  of  the  coefficients  a^,  a.^,  etc.,  are  zero,  the 
equation  is  said  to  be  complete  ;  and  if  one  or  more  of  them 
are  zeros,  incomiMe. 

Any  value  of  x  which  causes  the  first  member  of  (1)  to 
vanish,  or  become  zero,  is  called  a  root  of  the  equation. 

It  is  proved  in  Higher  Algebra  that  every  equation  of 
the  above  form  has  at  least  one  root,  and  we  shall  assume 
this  to  be  true  in  the  present  chapter. 

279.  Theorem  1.  If  a  is  a  root  of  the  equation 

x""  +  a^x''-'^.  +  «2^"~^  +  .  .  .  «„-i^  +  rt'n  =  0, 
the  first  rneinber  of  the  equation  is  divisible  hy  x  —  a. 


382  DEPRESSION  OF  EQUATIONS 

The  division  of  the  first  member  hy  x  —  a  may  be  con- 
tinued until  the  remainder  does  not  contain  x.  Denote 
this  remainder  by  R  and  the  quotient  obtained  by  Q.  Then 
we  have 

{x-a)Q^R  =  0, 

as  a  form  which  the  general  equation  may  be  made  to  as- 
sume. 

But  a  is  assumed  to  be  a  root  of  the  equation.  Hence 
if  we  put  X  —  a,  the  first  member  must  vanish. 

...  o.e  +  ^  =  o, 

or  i?  =  0. 

Therefore  x  —  a  \^  contained  in  the  first  member  with- 
out a  remainder. 

280.  Theorem  II.  Conversely,  if  the  first  me7)iber  of 
the  equation 

x^  +  a^x'^'^  +  a.iX^~'^  +  •  .  .  ctfi-i^  +  «„  =  0 

is  divisible  hy  x—  a,  then  a  is  a  root  of  the  equation. 

In  this  case  the  equation  may  be  made  to  take  the  form 

{x  —  a)Q—  0, 

the  first  member  of  which  vanishes  when  x  =  a.    Therefore 
a  must  be  a  root  of  the  equation. 

Cor.  If  the  first  member  of  the  equation 

be  divisible  by  ax  -\-  h,  then is  a  root  of  the  equation. 

281.  Theorem  III.  An  equation  of  the  nth  degree  has 
n  roots. 

We  have  assumed  what  may  be  proved  in  more  advanced 
algebra  that  the  equation 

x""  +  a^x'^''^  -\-  «2^""^  +  •  .  .  an-iX  +  «„  =  0 

has  at  least  one  root. 


DEPRESSION  OF  EQUATIONS.  383 

Denote  this  root  by  a.  Tlien  the  first  member  is  divisi- 
ble by  X  —  a,  aud  the  equation  may  be  written 

{x  -  «)(:r"-i  +  h^x""-^  +  .  .  .  K-yX-^  hn)  =  0, 

of  which  re  =  rt  is  a  solution,  and  of  which  a  farther  solu- 
tion may  be  obtained  by  putting 

a;"-i  +  b.x''-'  +  .  .  .  b^.^x  +  ^^  =  0. 

This  division  lowers,  or  depresses,  the  degree  of  the 
equation  by  unity.  The  new  equation  is  the  same  in  form 
as  (1),  and  therefore  may  be  assumed  to  have  at  least  one 
root. 

Denote  this  root  by  h.  Then  the  first  member  is  divisi- 
ble hy  X  —  h,  and  the  equation  may  be  written 

{x  -  h){x--'  +  c,x^-'  +  .  .  .  Cn-,x  +  c^)  =  0, 

of  which  a;  =  ^  is  a  solution,  and  of  which  a  further  solu- 
tion may  be  obtained  by  putting 

Jb  I      v^JC'  I      •    «    •    (yyi  —  i^     l~  t/^  —  \y» 

The  degree  of  this  equation  has  been  depressed  two 
units  from  that  of  (1).  It  is  still  of  the  same  general  form 
as  (1),  and  may  be  assumed  to  have  at  least  one  root. 

Denote  this  root  by  c.  As  the  first  member  is  divisible 
hy  X  —  c,  the  equation  may  be  written 

{X  -  C)(X^-'  +  chx^-'-i-   .   .    .   dn-iX  +  dn)  =  0, 

and  may  be  solved  by  putting 

X  —  c  =  0, 
and  x""-^  +  d.x''-''  +  .  .  .  dn-iX  -{-dn  =  0. 

The  degree  of  our  original  equation  has  been  depressed 
now  by  three  units. 

This  process  may  be  continued  till  the  degree  of  the 
original  equation  has  been  depressed  n  —  1  units,  and  we 
reach  an  equation  of  the  first  degree  of  the  form  x  —  k  =  0, 
of  which  k  is  the  root. 


384  DEPRESSION  OF  EQUATIONS. 

As  each  division  by  a  linear  factor  depresses  the  degree 
of  the  equation  by  unity,  it  must  be  divided  by  ^^  —  1  fac- 
tors to  depress  it  to  the  first  degree.  This  implies  n  —  1 
roots,  which  together  with  the  root  of  the  resulting  linear 
equation  make  n  roots. 

Cor.  1.  The  equation 


(a) 


^•n  _^  ^^^n-l  _^  ^^^n-2  _|_  ^ 

.  .  a^_,x  +  a„  =  0 

may  be  written 

{x  —  a){x  —  b){x  —  c)  .  . 

.  to  n  factors  =  0; 

and  the  equation 

AoX''  +  A,x''-'-{-A,x^-^-\-  .  . 

.  A^.,x  +  A,  =  0 

may  be  written 

Aq{x  —  a)(x  —  b){x  —  c)  .  . 

.  to  72  factors  =  0. 

(3) 

Cor.  2.  The  substitution  of  any  oth§r  than  one  of  the 
n  values  a,  d,  c,  etc.,  for  x  in  the  first  member  of  (2)  or 
(3)  would  not  cause  it  to  vanish.  Hence  an  equation  of 
the  ni\\  degree  has  only  7i  roots. 

Of  these  7i  roots  some  may  be  rational,  some  may  be 
surd,  and  some  may  be  imaginary.  Also  some  of  the  n 
roots  may  be  equal.  . 

Cor.  3.  The  solution  of  an  equation  of  the  ni\\  degree 
consists  merely  in  resolving  it  into  its  linear  factors,  and 
equating  each  of  these  factors  to  zero. 

Cor.  4.  The  degree  of  an  equation  in  x  may  be  de- 
pressed by  unity  by  dividing  it  through  by  x  minus  one  of 
its  roots. 

Cor.  5.  An  equation  in  x  may  be  tested  for  a  suspected 
root  by  dividing  it  through  by  x  minus  the  suspected  root. 

Cor.  6.  When  all  the  roots  but  two  of  an  equation  in 
X  are  known,  the  equation  may  be  depressed  to  a  quadratic 
equation,  which  may  then  be  solved  by  the  rule  already 
given. 


DEPRESSION  OF  EQUATIONS  385 

EXERCISE  CXLI. 

Form  the  equations  which  have  the  following  roots : 
1.     I,  2,  and  3.  2.     —  2,  —  3,  4,  and  5. 

3.     1,  -  2,  -  3,  and  0.  4.     4,  -  1,  -  3/2,  ^d  1/3. 

5.     -3,  -3,  4/3,  and  4/3.    6.     3,  -  4,  -  1/4,  and  1/5. 

Prove  that  the  numbers  given  are  roots  of  the  equation 
and  find  the  other  roots.     In  testing  for  suspected  roots, 
use  method  of  synthetic  division : 
Equation. 

7.  x^  -  ^Ix  +  84  =  0. 

8.  2a;3  +  bx^  -  4:3x  -  90  =  0. 

9.  x^-\-2x^ -nx-i-Q  =  0. 

10.  4:X^  -  4:X^  -7x^-4.x-{-4:  =  0. 

11.  9x^  -  Ux^  -  2a;2  _  24:c  +  9  =  0. 

12.  Sx^  -  Ux^  +  20a;  -  8  =  0. 

13.  x^  -  15.^2  +  10:r  +  24  =  0. 

14.  x^-4:X^-5x^+20x'^+4:X-li)=0. 

15.  a^-74:X^-24:X^-{-937x-S^0=0, 


Number. 

4. 

-  5. 

2. 

1/2,  2. 

1/3,  3. 

2/3. 

-1,2. 

1,  -  1,  2. 

1,  3,  -  5. 

CHAPTER  XXX. 
UNDETERMINED     COEFFICIENTS. 

A.       FUNCTIOI^S   OF   FINITE   DIMENSION'S. 

282.  Theorem  I.     A71  integral  expression  of  the  ntJi 
in  X  cannot  vanish  for  more  than  n  values  of  x,  ex- 
cept the  coefficie7its  of  all  the  poivers  of  x  are  zero. 

Let  Ax""  +  J5a:"-i  +  Cx''-^  +  .  .  . 

vanish  for  the  n  values  of  x,  a,  h,  c,  .  .  .     It  must  then 
be  equivalent  to  A{x  —  a){x  —  h){x  —  c)  .  ,  . 

If  now  we  substitute  for  x  any  value  k  different  from 
each  of  the  n  values  a,  b,  c,  .  .  .  ,  we  have 

A{k  -  a)(k  -  b){k  -  c) .  .  . 

Now  as  k  is  different  from  a,  b,  c,  .  .  .  ,  the  expression 
cannot  vanish  for  the  value  x  =  k,  except  A  itself  is  zero. 
If  A  be  2ero,  the  original  expression  reduces  to 

which  is  of  the  (fi  —  l)th  degree,  and  as  before  can  vanish 
for  only  71  —  1  values  of  x,  except  B  =  0.     And  so  on. 

Hence  an  expression  of  the  nth.  degree  in  x  catmot  van- 
ish for  more  tha7i  71  values  of  x,  except  the  coefficie7its  of  all 
the  powers  of  x  are  zero;  and  when  all  these  coefficients  are 
zero,  it  is  evident  that  the  expression  must  vanish  for  all 
the  powers  of  x. 

283.  Theorem  II.  If  ttvo  integral  expressions  of  the 
nth  degree  in  x  be  equal  to  one  another  for  more  than  7i 


PARTIAL  FRACTIONS.  387 

values  of  x,  tliey  ivill  be  equal  for  all  values  of  x,  and  all 
the  coefficients  of  the  saine  powers  of  x  in  the  two  expres- 
sions mMst  he  equal. 
Let 

Ax^^Bx""  -  ^^Cx^-^-{-  .  . .  =A'x''-\-B'x''-'^^C'x''-^-\- . . . 

Then  must     A  =  A\     B  =  B',     C^C  .  .  . 

By  transposition,  we  have 

{A  -  yl>'"  +  (^  -  B')x"-'  ^{C-  6'>«-2 .  .  .  =  0, 

and  this  must  be  true  for  all  values  of  x  for  which  the  two 
original  expressions  are  equal,  and  therefore  for  more  than 
n  values  of  x.     Hence  by  Theorem  I, 

A  -  A'  =  0,     B  -  B'  =0,     C-  C  =  0,  .  .  . 

or  A  =  A',  B  =  B',  C  =  C,  .  ., 

When  two  integral  expressions  in  x  of  finite  dimensions 
are  equal  for  all  values  of  x,  all  the  coefficients  of  the  same 
power  of  X  in  the  two  expressions  must  be  equal  to  each 
other.  For  in  this  case  n  is  finite,  and  the  possible  values 
of  X  infinite,  and  therefore  >  n. 

B.       PARTIAL    FEACTIONS. 

284.  Definition  of  Partial  Fractions.— The  sum  of  the 

two  fractions  ^ and  - — ■ —  is 


1  —  X  l-\-  X      1  —  x^ 

With  reference  to  the  last  fraction,  the  parts  which 
make  it  up  by  addition  are  called  its  partial  fractions.  It 
is  often  necessary  to  separate  a  fraction  into  its  partials. 
In  this  separation  it  is  understood  that  the  denominators  of 
the  partials  shall  be  of  the  first  degree  when  practicable, 
but  at  any  rate  of  a  lower  degree  than  that  of  the  original 
fraction. 

e.g.     1.   Separate  ^  into  partial  fractions. 

x         X 


388  UNDETERMINED   COEFFICIENTS. 


Since  the  denominator  =  {1  —  x){l  -\-  x),  assume 


2  + 8a;  A      ^       B 


+ 


l-x^  ~  1  -  x^  1  -\-  x' 

in  which  A  and  B  are  coefficients  to  be  determined. 
Clearing  of  fractions,  we  have 

2  +  8a;  =  ^(1  +  a;)  +  B{1  -  x) 

=  (^  +  B)x'^  +  (^  -  B)x. 

And  as  this  is  to  be  true  for  all  values  of  x,  we  may 
apply  Theorem  II,  which  gives 

^  +  ^  =  2, 

and  A-  B^S. 

.'.     2.4  =  10,     and    A  =  b. 

Also,  2^  ==  —  6,  and     ^  =  —  3. 

Hence  the  partials  are 

and     —  :; — ■ — . 

1  —  X  1  -\-x 

From  the  above  example  we  may  derive  the  following 
rule  for  separating  a  prope'r  fraction  into  its  partials: 

Resolve  the  denominator,  if  possible,  into  real  linear 
factors,  and  form  fractions  icith  undetermined  numerators, 
and  put  their  sum  equal  to  the  original  fraction.  Clear  of 
fractions,  and  equate  the  coefficiefits  of  the  like  powers  of  x. 

EXERCISE  CXLII. 

Separate  the  following  fractions  into  partials  with 
linear  denominators : 


1. 


7x  +17  34  -  2x 


x^-\-5x-j-  6*  x^  +  2a;  -  8 

25  -  a;  13a;  -  26 


3.      -0 T^-  4. 


x^  -  X  -  12'  a;2  _  3a;  -  40 


PARTIAL  FRACTIONS. 


389 


llx  -  7 


'■^x'  -lx-16' 


c    c.  .  a;^  +  3a;  +  2 

e.g.    2.  Separate  ^(^  _  ^^^^ 

fractious. 


10  -  15a; 

6a:2-26a;+24' 

into  partial 


Assume 

a;2  +  3^  +  2 


A 


2)(x-3) 


5 


a;-  2 


3* 


6(a;  -  l)(a;  -  2)(a:  -  3)  ~  Q{x  -  1) 

Theu,  cleariug  of  fractions,  we  have 

a;2  +  3a;  +  2 
=  ^(x-2)(a;-3)+65(a;-l)(a;-3)+6(7(a;-l)(a;-2) 

=  Ax^-bAx^QA-^QBx'-UBx-^rl^B^QCx^-l^Cx-{-l'ZG 

x^ 


=  A 

^ 

-    6A 

x+    QA 

-\-QB 

-24:B 

+  18^ 

+6(7 

-ISC 

+  126' 

Therefore,  equating  coefficients,  we  have 
A-\-    6B-{-    6(7=1, 
5A  +  24^  +  18(7=  -  3, 
and  6.4  +  18i?  +  126'=2. 

Whence         A  =  3,   B  =  -  2,  and  C=  5/3. 

a;2  +  3a;  +  2  1  2  5 


•  *     6(a;-l)(a;-2)(a;-3)      2(a;-l)       a;  -  2  ^  3(a;  -  3)  * 

There  is,   however,  a  shorter  way  of  solving  this  ex- 
ample.    Since  in  the  expression 

a;2  +  dx  -\-2  =  A{x-  2){x  -  3) 

+  QB{x  -  l)(a;  -  3)  +  6(7(a;  -  l)(a;  -  2) 
X  may  have  any  value  whatever,  we  may  put  a;  =  1. 


390  UNDETERMINED  COEFFICIENTS. 

Then  we  shall  have 

6  =  2.4,     and     ^  =  3. 
If  we  put  X  =  ^,  we  shall  have 

12  =  -  Q>B,     and     B=  ~  2. 
If  we  put  X  =  3,  we  shall  have 

20  =  12(7,     and     C  =  5/3. 

It  is  much  shorter  to  use  this  method  when  by  inspec- 
tion we  can  find  values  of  x  which  will  cause  all  the  terms 
except  one  of  the  right-hand  member  of  the  identity  to 
vanish. 

EXERCISE    CXLIII. 

Separate  the  following  fractions  into  their  partials : 

^2  -  Ux  +  37  9^2  _  3g^  _  g9 

2. 


"•    (x-3){x^-9x-\-20Y  {2x-j-2)(x^-9y 

2dx  -  Ux^  3x  -  2 

3.      77i TwT^ 57-  4. 


(2x-l)(9-x'y  "•    (x-l){x^-  5x-{-Qy 

X  2;^  J-  X  +  1 

6.  ' 


*"    (x  +  l)(x  +  3){x -j- 5y  "•    (x-]-l)(x^-5x-^Qy 

7a;2  _|_  7^^  _  6 

e.g.     3.  Separate  -, — r~rW o\  ^^^^  ^^^  partials. 

(.T+  1)  (a;  —  Z) 

In  forming  this  fraction  by  addition  there  may  have 

A 

been  a  fraction  in  the  form  of  -, — t^tto,  one  in  the  form  of 

(x-j-iy 

7?  n 

and  one  in  the  form  of -.      Hence  in   our  as- 


x+r  x-2 

sumption  we  must  make  provision  for  all  these. 

7a;2  +  7a;-6              A        ^      B      ^       C 
Assume  -. — r^rY27 ^  —  i  — r--i  \2  H r^  + 


{x^\)\x-%)      {x^Vf'  x-\-\'  x-^^ 


PARTIAL  FRACTIONS.  801 

Clearing  of  fractions,  we  have 

Ix'^lx  -  6  =  A{x  -  2)  +  ^(x+  l)(a:-2)+  C(^+l)l 

-  Putting  X  —  —  I,  we  have 

-  6  =  —  3^,    and     A  =  2. 

Putting  :c  —  2,  we  have 

36  =  9C,    and     (7=4. 

Equating  coefficients  of  x^,  we  have 

^+  (7=7. 

.-.     ^=  7  -  C=3. 

7a:^  +  7a;  -  6     _        2  3  4 

^®^^®  (2:  +  1)2(:?;  -  2)  ~  {x^lf  +a;  +  l  +  2;-2- 

e.g.     4.  Separate  yg —  into  partials. 

X  J. 

The  denominator  =  {x  —  l){x^  -\-  x-\- 1)^  and  the  qua- 
dratic factor  is  not  separable  into  real  factors. 

But  a  proper  fraction  which  has  a  quadratic  factor  for 
its  denominator  may  have  a  linear  factor  for  its  numerator. 
We  must  make  provision  for  this  by  assuming  that 

bx^  +  l         A  Bx-\-C 


x^  —  1       x  —  1      x^  -\-  x-\-  1' 
6x^-^1^  A{x^  +  x-\-l)-^{Bx-\-  C){x 
Putting  ic  =  1,  we  have 

6  =  3A,    and     ^  =  2. 

Equating  the  coefficients  of  x^,  we  have 

A-\-B  =  6. 

...     5  =  5-^  =  3. 

Equating  the  constant  terms,  we  have 

A+  C=l. 


392  UNDETERMINED   COEFFICIENTS. 

Whence  C  =  1  --  2  =  -  1. 

bx'  4-1  2       ,        'dx  -  \ 


Therefore 


x^  —  1        X  —  1       x^  -{-  X  -\-  1' 

Observe  that  each  of  the  separations  into  partial 
fractions  given  is  characterized  by  this :  that  it  introduces 
just  as  many  undetermined  coefficients  as  equations  for 
them  to  satisfy.  This  is  characteristic  of  any  proper 
application  of  the  method  of  undetermined  coefficients  in 
which  the  number  of  coefficients  is  finite. 

EXERCISE  CXLIV. 

Separate  the  following  fractions  into  partials: 

12a;2  -  a:  +  10 


1. 


6. 


1 

x^  +  r 

2a;3  +  2x^  -f  10 

x'-i-: 
X^  - 

-X+1 

4. 


x^-1 

x^  -  d 


(^  +  m^''  + 1)' 


(x^  -]-i){x  -  ly 

G.    FUNCTIONS   OF   INFINITE    DIMENSIONS. 

285.  Theorem  II.  If  Uvo  integral  functions  of  x  of 
infinite  dimeyisions,  and  arranged  in  asce^iding  order,  are 
equal  to  one  another  for  all  values  of  x  ivUich  malce  the 
series  convergent,  the  coefficients  of  the  like  powers  of  x  in 
the  two  series  will  he  equal. 

JjQt  A^hx-{-  Cx^^  .  .  ,  =  A'  ^  B'x  +  C'x^  +  .  .  . 
be  true  for  all  values  of  x  which  render  both  convergent. 

Then  will     A=:  A',     B  =  B',     C  —  C ,  etc. 

For  if  the  series  are  both  convergent  their  difference 
will  be  convergent,  and  we  shall  have 

A-  A  ^{B-  B')x  +  (C  -  C')x^  .  .  .  =  0 
for  all  values  of  x  for  which  the  series  is  convergent. 


EXPANSION  OF  FUNCTIONS.  393 

But  when  x  is  sufficiently  small,  the  series  is  convergent 
and  A  —  J'  is  greater  than  all  that  follows,  and  its  sign 
must  control  that  of  the  series;  that  is,  the  A  —  A'  will  be 
>,  =,  or  <  zero  according  as  the  series  is  >,  =,  or  < 
zero.     But  the  whole  series  =  0. 

.-.     A-A'  =  0,     or    A  =  A'. 

By  striking  out  A  and  A'  as  equal,  we  may  in  like 
manner  prove  B  —  B' ;  and  then  C  =  C,  etc.     For  since 

{B  -  B')x  +  {C  -  C')x  4-  .  .  .  r=  0 

for  all  values  of  x  which  make  the  original  series  convergent, 
and  therefore  for  other  values  of  x  than  zero,  both  members 
of  the  equation  may  be  divided  by  x  and  the  conclusion  be 
drawn  that 

B  -  B'  -^{C-  C')x^.  .  .  =:0 

for  values  of  :r  which  make  the  original  equation  convergent. 

D.       EXPANSION    OF   FUNCTIONS. 

A  function  may  be  developed  into  an  infinite  series  in 
various  ways;  and  whenever  the  series  is  convergent,  the 
function  is  equal  to  its  development,  which  is  then  called 
its  expansion.  It  is  important  to  bear  in  mind  that  when 
the  series  into  which  a  finite  function  is  developed  becomes 
divergent  for  any  value  of  x  the  function  cannot  equal  its 
development. 

A  proper  fraction  may  be  developed  into  an  infinite 
series  in  ascending  powers  of  x  by  division. 

The  four  following  expansions  by  division  are  im- 
portant : 

1.  -^—  =  1  +  a;  +  2^2  +  ic3  +  a;^  -f  .  .  . 
1  —  x 

2.  -4— =1-^  +  ^^-^^+^;^+.  .  . 
l-\-x 


894 


UNDETEBMINED  COEFFICIENTS. 


3. 


4. 


(1  —  a;) 
1 


5x^ 


{1  +  xy 


l-2x-\-3x^-  4:x^  +  5a;^  +  . .  . 


A  function  which  is  not  a  perfect  power  may  be  devel- 
oped into  an  infinite  series  in  ascending  order  by  evolution. 


e.g. 


yi  -  :c  =  1 


16 


6x^ 
128'  •  • 


If  a  function  of  x  which  has  but  one  value  for  each 
value  of  X  be  expanded  in  ascending  powers  of  x,  the  powers 
must  all  be  integral. 

For  were  the  exponent  of  any  term  to  become  fractional, 
that  term  would  be  many-valued  for  eacli  value  of  x,  which 
contradicts  the  hypothesis. 

The  following  example  illustrates  the  expansion  of  a 
fraction  by  the  method  of  undetermined  coefficients. 

Expand  — — — — -^  to  five  terms  in  ascending  powers 

JL  ""j~  X  ~j~  X 

of  X. 

Assume 

1  2 

~^~^2  =  A-i-Bx+  Ct?  4-  B^x^^Ex^'-^Fx^^  Qx^-^.., 

1  -  re  -  a;2  =  ^(1  +  a;  +  a;2)  +  B{x  ■\- x^  ^  x^) 
-f  C{x^  +  a;3  +  x"-)  +  D{x?  +  rc^  +  x"")  +  E{x^  +  ^'  +  ^') 
+  Fix''  +  a:«  +  x')  +  G{x'>  +  a;"^  +  a:8)  +  .  .  . 


A^  A 


x+  A 

7?+B 

x'+C 

vf'  ^D 

^+E 

-\-B 

+  0 

+  D 

■\-E 

+  F 

+  c 

+  D 

+  E 

^F 

+  G 

X^-\- 


Whence 


EXPANSION  OF  FUNCTIONS.  395 


A-\-B=  -1, 

and 

B=-2, 

A+B+C=-h 

and 

C  =  0, 

B  +  C-\-  D  ^0, 

and 

i)  =  2, 

(7  +  i>  +  ^  =  0, 

and 

E  =  -2, 

D  +  E-\-  F  =0, 

and 

F=0, 

E-{.  F-\-  G  =  0, 

and 

G  =  2. 

;+:+:.-!  ^-+^-^- 

-  2x^  +  2x^  + 

and 


In  certain  cases  the  operation  of  expanding  fractions 
into  series  may  be  abridged. 

1°.  If  the  numerator  and  denominator  of  the  fraction 
contain  only  even  powers  of  x,  we  may  assume  a  series  con- 
taining only  even  powers,  as  ^  +  Bx^  -\-  Cx'^  +  .  •  . 

2°.  If  the  numerator  of  the  fraction  contains  only  odd 
powers  of  x  and  the  denominator  only  even  powers,  we 
may  assume  a  series  containing  only  odd  powers  of  x. 

3°.  If  every  term  in  the  numerator  contains  x, 
but  not  every  term  in  the  denominator,  we  may  assume 
a  series  beginning  with  the  lowest  power  of  x  in  the 
numerator. 

4°.  If  the  numerator  does  not  contain  x,  we  may  find 
by  actual  division  what  power  of  x  will  occur  in  the  first 
term  of  the  expansion. 

e.g.       ^  _ — 3  gives  by  division  l/3a;~*  as  the  first  term 

of  the  quotient.     Hence  we  may  assume 


396  UNDETERMINED  COEFFICIENTS. 

EXERCISE  CXLV. 

Expand  each  of  the  following  fractions  to  five  terms  in 
ascending  powers  of  x : 

l-%x-\-  Zx^ 


3. 


1  +  32;  -  ^X^' 

3    -    4:0? 


4. 


dx 

^-        4:-dx'' 


2 

-  3a:  +  4a;2 

1 

+  2x  - 

-  5x' 

2 

-  ^x? 

1 

+    4:X^' 

2x 

3  -  2x^' 


The  following  example  illustrates  the  method  of  develop- 
ing a  radical  by  the  method  of  undetermined  coefficients. 
To  expand  Vl  +  ^• 
Assume 


Vl  -\-  X  =  A-^Bx  +  Cx^-i-  Dx^  +  Ex^  + 
Then,  squaring  each  member,  we  have 


l-\-x=:A''  +  2ABx  -i-2AC 
-\-B^ 


Whence 


x^-\-2AD 
+  2BC 


x^-{-2AB 
-{-2BD 


x'  + 


A^  =  1,  A  =  1, 

2AB  =  1,  B  =  1/2, 

2AC+B^=:0,  6'=  -1/8. 

2AD-\-2BC=:0,  D  =  l/16, 

2AE  +  2BD  +  C2  =  0,  B=-  5/138. 
Therefore 
^(l-\-x)  =  l  +  \/%x  -  l/W  4-  l/16a;3  -  5/128:^4+  . . 


EXPANSION  OF  FUNCTIONS. 
EXERCISE  CXLVI. 

1.  Expand  1/(1  +  x  +  x^)  to  x\ 

2.  Expand  y  (———)  to  x^. 

3.  Expand   \/{l  +  x)  to  x^. 

Ex.      Let  ij  =  dx-2x^-\-  3x^  -  4:X^ -{-  .  ,  , 
Express  x  in  ascending  powers  of  2/  to  ?/*. 
Assume  x  =  Ay  -{-  By'^  -\-  Cy^  +  Dy^  -f  •  •  • 
=  A{^x-  2x^  +  3.?;3  -  4:X*  +  .  .  .^ 
+  ^(9^2  ~  12x^  +  22:^^  +  .  •  •) 
+  C{27x^-  54^-4 -f.  ..) 
+  i>(81a:* +  ..'.)• 


397 


.-.     x  =  3Ax-2A 

x^  +  3.4 

x^  -  4.4 

^^  +  .., 

+  95 

-125 

+  225 

+  27C 

-  54C 

+  S1D 

Whence                                          SA  =  1 

-2^  +  95  =  0 

3J  -125  +  27(7=0 

-  4^  +  22 5  -  54C  +  81i>  =  0. 

Whence 

A  =  1/3,  ^  =  2/27,  C*=  -  1/243,  D=-  14/2187. 

Therefore  x  =  l/'dy  +  2/2? 

f-1/2 

43?/3-l4 

/21873/^+.. 

398  UNDETERMINED   COEFFICIENTS. 

EXERCISE  CXLVII. 

1.  It  y  =  2x  -\-  x'^  —  2a^  —  3x^  -{-...,  find  x  in  terms 
of  y  to  y^. 

2.  It  y  :=  X  -\-  x^  -\-  x^  -\-  x^  -\-  ,  .  .  ,  find  x  in  terms  of 
y  to  y\ 

3.  lty  =  x  —  a^-{-x^  —  x'^-\-,..,  find  x  in  terms  of 
y  to  y\ 


CHAPTEK  XXXI. 
CONTINUED  FRACTIONS. 


286.  Definition  of  a  Continued  Fraction. — An  expres- 
sion of  the  form 


a± 


o±  — 


g  ±  etc. 

is  called  a  contmued  fraction. 

For  convenience,  continued  fractions  usually  are  written 
in  the  form 

a  ±  -     -     —      etc. 
c  ±e  ±g  ± 

In  this  chapter  we  shall  consider  only  the  simpler  form 
«,  H ,  —  ,    etc., 

in  which  the  numerators  are  each  unity  and  «, ,  a^,  a^, 
etc.,  are  positive  integers. 

The  fractions  a, ,  ^,  — ,  etc.,  are  called  the  first,  sec- 

ond,  third,  etc.,  elements  of  the  continued  fraction. 

287.  The  Convergents. — The  fraction  obtained  by  stop- 
ping at  any  element  is  called  a  convergent  of  the  continued 

fraction.    Thus  a^ ,  a.  -\ ,  and  a,-\ ,  —  are  the  first, 

«,  a^  +«3 


400  CONTINUED  FRACTIONS. 

second,  and  third  convergents  of  the  continued  fraction 
given  above.     These  convergents  may  be  reduced  to  the 

forms  f',  «-^^-±i,  and  <"■"'  + ^j"' +  "■. 
1  <?„  a^a„  4-  1 


For,  evidently,  a^  =  ^, 


1  _  a,a^       1   _  a,a^  +  1 

Q/.  -\-  —  -\-  —  y 

a^  «2  «2  «a 

and    «,  H —  a^-\ —~  —  a,  -\ ^ 

The  rth  convergent  of  a  continued  fraction  will  be  de- 
noted by  — . 

Each  convergent  may  be  reduced  to  an  ordinary  frac- 
tion, as  above,  by  successive  simplification  of  the  complex 
fractions  of  which  it  is  composed.  In  this  simplification 
we  begin  always  with  the  last  complex  denominator. 

288.  Theorem  I.  The  numerator  and  denominator 
of  any  convergent  beyond  the  second  are  formed  by  nuilti- 
plying  the  numerator  and  deno7ninator  of  the  iweceding 
convergent  by  the  denominator  of  the  new  element  considered 
and  adding  to  the  respective  products  the  numerator  and 
denominator  of  the  last  convergent  but  one. 

An  examination  of  the  first  three  convergents  already 
obtained  by  actual  reduction  of  the  complex  fractions  to 
simpler  ones  will  show  that  the  numerator  and  denominator 
of  the  third  convergent  are  formed  in  accordance  with  this 
theorem. 

Denote  the  number  of  the  convergent  by  n,  and  the  nth 


CONTINUED  FRACTIONS.  401 


convergent  by  — ,  the  preceding  convergent  by  — — ^ ,  the 

Qn  Qn-l 

last  but  one  by  ^^^^'. 

Qn-2 

Then  in  the  case  of  the  third  convergent  we  have 

Now  each  convergent  differs  from  the  one  preceding  it 
by  having  an  -\ substituted  in  place  of  a„.     Thus  the 

^n  +  1 

second  convergent  differs  from  the  first  simply  in  having 
a^-\ in  place  of  «, ,  the  third  differs  from  the  second 

in  having  a^-\ in  place  of  a^ ,  and  the  (n  -\-l)^i  will 

differ  from  the  n  only  in  having  a^  -\ in  place  of  a„. 

Making  this  change  in  (1),  we  have 


Pn-ir  1  \ ^H  +  1 ' 


_Q?n+l   {anVn-^-\-Pn-%  -\-)  Pn  -  1  _(tn  +  I  Pn  +  Pn  -  1 
0^«+rK^n-l+5'n-2+)5'n-l        «n  +  1  5'n  +  ^n  -  1  ' 

which  agrees  with  the  theorem. 

Hence  the  theorem  which  holds  for  the  third  convergent 
holds  also  for  the  fourth,  the  fifth,  and  each  subsequent 
convergent. 

Therefore  the  formula  for  the  rth  convergent  is 

qr  ttrqr-l-\-  qr-2 


402  CONTINUED   FRACTIONS. 

289.  Partial  and  Complete  Quotients. — The  integers 
a,,  «2,  «3,  etc.,  may  be  called  the />rt?-^{a/ quotients,  «5„  being 
the  nth  partial  quotient.  When  the  number  of  partial 
quotients  is  finite  the  continued  fraction  is  said  to  be 
termi7iating .  If  the  number  of  these  quotients  is  un- 
limited the  fraction  is  called  an  infinite  continued  fractio7i. 

Since  a^,  a^,  a^,  etc.,  are  positive  integers,  a  continued 

fraction  of  the  form  a,  -\ , 1-  etc.  must  be  greater 

«^,  +  «,  ^ 

than  unity;   while  a  continued  fraction   of   the  form   of 

—  -L  —      —  must  be  less  than  unity. 

«,     ^      «,     +    ^3      +    .     .     .  ^  ^ 

The  complete  quotient  at  any  stage  is  the  quotient  from 
that  point  on  to  the  end.      Thus  an  is  the  nth  partial 

quotient,  and  a„  -\ is  the  correspond- 

^n  +  1      I      ^n  +  2      I     •   •  • 

ing   complete  quotient.     The   complete   quotient   at   any 
stage  may  be  denoted  by  K. 

As  we  have  seen,  the  ^th  convergent  is 

Qn  a„qn  -  1  +  5'n  -  2  ' 

This  value  evidently  may  be  converted  into  that  of  the 
whole  continued  fraction  by  substituting  K  in  the  place  of 
cin-    Denote  the  value  of  the  entire  fraction  by  x.    Then  will 

Kq„  _  1  +  (7„  _  2  * 

290.  Theorem  II.  The  difference  between  tivo  suc- 
cessive convergents  is  a  fraction  whose  numerator  is  imity 
and  whose  denomi^iator  is  the  product  of  the  denominators 
of  the  convergents,  and  this  difference  taken  in  regular 
order  is  alternately  positive  and  negative. 

Pn^  Pn-l    _(fnP„-l+P,,-2  Pn  -  1 


qn      qn-i      «uqn-i-\-  qn-2       qn-i 


CONTINUED  FRACTIONS.  403 

~  {Clnqn-l+  qn-'z)gn-l 

Pn_Pn-l    _Pn-2qn-l—Pn-  l^n  -  2 
qn  qn-l~  qnqn-1 

.*.      Pnqn-l-Pn-iqn  =   "  (i?n  -  l5'n  -  2  "  i?n- 25'n  -  l)- 

So  also  in  succession 

Pn-iqn-2  -  Pn-2qn-l=  -  i?n  -  25'n  -  3  +  i?n  -  sS'n  -  2- 

p.q.-p.qz  =  -p.q.+p^q.' 

But    p,q,  -  p,q,  =  (a^a,  +  1)  -  a^if,  =  1  =  ("  1)^- 

Also,  since  the  successive  convergents,  beginning  with 
the  first,  are  alternately  less  and  greater  than  the  fraction, 
the  successive  convergents  are  alternately  greater  and  less 
than  the  preceding.  Therefore  the  successive  difference 
will  be  alternately  positive  and  negative,  so  that  the  numer- 
ator of  the  fraction  will  be  (—  1)",  in  which  n  is  the  num- 
ber of  the  convergent  used  as  a  subtrahend. 

Hence        Pnqn-i  -Pn-iqn  =  (-  1)".  (1) 

Hence,  also,    ^  -^^^  =  till!.  (2) 

q,t     qn-i      qnqn-i  ^ 

CoK.  1.  All  convergents  are  in  their  lowest  terms. 
For  every  common  measure  of  jt7„  and  q^  must  also  be  a 
measure  of  Pnqn-\  —  Pn-\qn  and,  from  (1),  of  ±  1. 

Hence  pn  and  q,^  can  have  no  common  measure. 

Cor.  2.     In  the  continued  fraction 

_i-      -i-      _1_ 

«,     +    a,     -f     ^3+ ' 


404  CONTINUED  FRACTIONS. 

wliicli  is  less  than  unity, 

^, /y        _^       n—(  —  ^Y-'^    Olid   -^"  _  Pn-\  _  {—  1)" 

Yn  Y«  -  1  7nYn  -  1 

since  the  first  convergent  will  be  too  large,  the  next  too 
small,  etc. 

291.  Theorem  III.      Each   co^ivergent  is   nearer  in 
value  to  the  continued  fraction  than  any  preceding  con- 


Let  X  denote  the  continued  fraction,  and  — ,  ^C!L±i 

qn     qn+i 

and  i-!?-±l  denote  three  consecutive  convergents. 

Then  x  differs  from  ^"^  ^  only  in  taking  the  complete 

5'ji+  2 

{n  -\-  '^)  quotient  in  place  of  «„  +  ^.     Hence 


X  = 


Kqn^x-^  qn  ' 


Pn      K{p^^^ q^  ~  p,,q^ -f  1 )  _  K 


qn{  Kqn  +  1  +  <Zn)  !Zn(  A'^„  +  j  +  q^f 

and 

qn+1  '"        qn+1  '^  J^qn  +  i+qn 

=    Pn+iqn'^  Pnqn  +  1     ^  1 

?n  +  l{J^^qn  +  1  +  ^»)  5'«  +  l(^^'^n  +  1  +  5'n)  ' 

Now  A"  >  1     and     q,,  <  <7„+i; 

hence  on  both  accounts 

K  .  1 


qnJi^qn  +  1  +  5'n         ^'n  +  i  A^„  +!+$'«* 

Combining  the  result  of  this  article  with  that  of  article 
290,  it  follows  that 


CONTINUED  FRACTIONS.  405 

The  co7ivergents  of  an  odd  order  continually  increase, 
hut  are  akuays  less  than  the  continued  fraction  j 

The  convergents  of  an  even  order  continually  decrease, 
but  are  always  greater  than  the  continued  fract\on. 

292.  Theorem  IV.     The  value  of  x  differs  from  — 

1         ,  ,  .,  1 


by  less  than  — ^  and  by  more  than      ^ 

Let  — ,    ^^"^  S        ^  ^    be    three    consecutive   conver- 
gents,  and  let  K  denote  the  {n  +  2)th  complete  quotient. 
Then  ^  =  pL±l±^n^ 

qn  {Kqn  +  1  +  qn)qn  qn(Kqn  +  1  +  ^n) 

_    Kp^  ^  iqn  +  Pnqn  -   ^^Mn  +  1   -  Pnqn 
qn{Kqn^X-\-qn) 

-    -^(Pn  +  iqn  -  Paqn.l)   _  ^^ 

qn{Kq^  f  1  +   ^n)  qn{Kqn  +  1  +  ?n) 

1 


4^„  +  l    +    |) 


Now  K  is  greater  than   1,  therefore  —  differs  from  2: 

qn 

by  less  than and  by  more  than — -5. 

Mn+.  qnqn-tl  +  qn 

And  since  q^  <  §'„  + 1 ,  the  difference  between  —  and  x 

qn 

must  be  less  than  -^  and  greater  than  ^-^ — . 

qn  <'q  n  +  l 

293.  Theorem  V.  The  last  convergent  preceding  a 
large  partial  quotient  is  a  close  approximation  to  the  value 
of  the  fraction. 


406  CONTINUED  FRACTIONS. 


By  the  last  theorem,  the  error  in  taking  -—  instead  of 

fin 

the  whole  continued  fraction  is  less  than  ,  or,  since 

(7n  +  i  =  «n  +  ign  +  ^n-i,   less  than  —y- -— — ^,  or 

less  than  j.     Hence  the  larger  a^  +i  is,  the  nearer 

does  —  approximate  to  the  continued  fraction.  Therefore 
when  a^  +  x  is  relatively  large,  the  value  of  x  differs  but 
little  from  that  of  ^. 

qn 

294.  Theorem  VI.  Every  fraction  whose  numerator 
and  denominator  are  positive  integers  can  he  converted  into 
a  terminating  continued  fraction. 

m 
Let  —  be  a  fraction  whose  numerator  and  denominator 

71 

are  positive  integers. 

Divide  mhj  n  and  let  a^  be  the  integral  quotient  and 
p  the  remainder.     Then 

m  .    p  ,1 

n  n         ^        n 

P 

Divide  n  hy  p  and  let  a„  be  the  integral  quotient  and  g 
be  the  remainder.     Then 

n  q  1 

P  ^  p  P 

q 

Divide  p^y  q  and  let  ^g  be  the  integral  quotient  and  r 
be  the  remainder,  and  so  on. 

Therefore  —  =  «,  H ,    — 

^  ^«a  4-  «3  +  •  •  • 


CONTINUED  m ACTIONS. 


407 


If  7)1  <  71,  the  first  integral  quotient  will  be  zero. 

w  1 

Put  —  = and  proceed  as  before. 

71  n 

m 

The  above  process  is  the  same  as  that  of  finding  the 
greatest  common  measure  of  m  and  n,  a^,  a^,  a^  being  the 
successive  quotients.  As  m  and  }i,  being  positive  integers, 
are  commensurable,  the  process  must  terminate  after  a 
finite  number  of  divisions. 


K7n 


Cor.     Evidently  —  and  -^r^-  will  give  the  same  contin- 
•^  71  Rn         * 

ued  fraction. 


e.g. 


251 
1.   Reduce  — —  to  a  continued  fraction. 

oO/C 


Find  the  greatest  common  divisor  of  251  and  802  by  the 
usual  method. 


quotients. 


251       1       1       1      1 

•'•     802  "3+5+8  +6* 

e.g.     2.  Reduce  3.1416  to  a  continued  fraction. 

1416 


251 

802 

3    ] 

6 

49 

5 

1 

8 
6 

3.1416  =  3  + 


10000 


1416 

8 


10000 

88 


7 
16 
11 


and 


3.1416  =  3  + 


1416 
10000 

1      2       i 

7 +  16 +  11* 


1      L      i 

7  +  16  +  ri' 


408 


CONTINUED  FRACTIONS. 


355 


e.g.     3.  Show  that  -— ^  is  a  close   approximation   to 
3.14159,  differing  from  it  by  less  than  .000004. 


3.14159  =  3  + 


159 

100000 

7 

854 

887 

15 

29 

33 

1 

1 

4 

25 
1 

7 
4 

1 

1   1 

1 

1 


7  + 15 -hl  + 25 +  1  +  7 +4 
The  successive  convergents  are 


3 
1' 


22 

r 


333      355 
106'    lis' 


The  last  convergent  precedes  the  large  quotient  25,  and 
hence  is  a  close  approximation  to  x. 

It  differs  from  it  by  less  than  — .       .„ ,  and  there- 


fore bv  less  than 


25  X  (100)' 


25  X  (113)2 
,  or  .000004. 


EXERCISE    CXLVIII. 

Express  the  following  as  continued  fractions; 

3.     3.61. 
144 


53 

72 

59* 

'      91- 

112 

749 

153' 

^'    326- 

436 

3015 

345* 

*•  6961 

6. 


89 


CONTINUED  FRACTIONS.  409 

Calculate  the  successive  convergents  to  the  following 
continued  fraction : 

24--      -      -      —      - 
^'        "^6+1+1  +  11+2' 

1  L      1       ?L      1        1 

^^'     2+2  +3  +1  +2  +  6* 

"•        "^3  +  1+2+2+1+9* 
1111 

12        -_        

2  +  3  +  1  +  4  * 

13.  Find  a  series  of  fractions  converging  to  .24226,  the 
excess  in  days  of  the  tropical  year  over  365  days. 

14.  A  metre  is  39.37079  inches;  show  by  the  theory  of 
continued  fractions  that  32  metres  are  nearly  equal  to  35 
yards. 

16.    A  kilometre  is  very  nearly  equal  to  .62138  mile. 

Qi,       .1.  ^  XT-    i.      ..        5    18    23      64 
bnow  that  the  tractions  — ,  — ,  ^r^,  -p—  are  successive  ap- 
proximations to  the  ratio  of  a  kilometre  to  a  mile. 

16.  Two  scales  of  equal  lengths  are  divided  into  162 
and  209  equal  parts  respectively.  If  their  zero  points  are 
coincident,  show  that  the  thirty-first  division  of  one  nearly 
coincides  with  the  fortieth  of  the  other. 

17.  The  modulus  of  the  common  system  of  logarithms 
is  approximately  equal  to  .43429.  Express  this  decimal  as 
a  continued  fraction,  find  its  sixth  convergent,  and  deter- 
mine the  limits  to  the  error  made  in  taking  this  convergent 
for  the  fraction  itself. 

18.  The  base  of  the  Napierean  system  of  logarithms  is 
2.7183  approximately.  Express  this  decimal  as  a  continued 
fraction,  find  its  eighth  convergent,  and  determine  the 
limits  to  the  error  made  in  taking  this  convergent  for  the 
fraction  itself. 


410  CONTINUED  FRACTIONS. 

295.  Periodic  Continued  Fractions. — AVlien  the  partial 
quotients  of  a  continued  fraction  continually  recur  in  the  same 
order,  the  fraction  is  called  2i  periodic  continued  fraction. 

A  periodic  continued  fraction  is  said  to  be  simple  or 
mixed  according  as  the  recurrence  begins  at  the  beginning 
or  not.     Thus, 

1      1,     1       1      1 
^"^^>+c+a  +  i^c  +.  .. 

is  a  simple  periodic  fraction. 
L      1     1     L 

is  a  mixed  periodic  fraction. 

296.  Theorem  VII.  A  quadratic  surd  can  he  ex- 
pressed as  an  infinite  periodic  continued  fractio7i. 

e.g.     Eeduce  V%  to  a  continued  fraction. 
The  integer  next  below    VSi^  2.     Hence 

1^8  —  2  expressed  as  an  equivalent  fraction  with  a 
rational  numerator  is 

(|/8-2)(l^+2)_         4 


y8  +  2  i/8  +  2 

V8  =  2  +  -— ^ =  2+        ^ 


4^+2  V84-2 


The  integer  next  below ^^—  is  1. 

V8  +  2  _  i^8- 2 


Hence 

1  1 


i^S  =  2  +  — — — =  2  + 


1        ^8-^  1+4/8 


CONTINUED  FRACTIONS.  411 

1 

=  2  +  i      (  V8  -  2)(  V8  +  2) 


!■  +  |/8  +  2 
The  integer  next  below  V8  +  ^  is  4.     Hence 

(l/8-2)(  V8 +3) 


4/8  +  2=4+4/8-2  =  4-1- 


^8  +  2 


4 

4+     ^      ^  =4  + 


V¥+2  4^8  +  2  * 

4 

At  this  point  the  steps  begin  to  recur : 

^=^  +  r+4  +  r+r+... 

Thus  4^8  is  seen  to  be  equivalent  to  a  periodic  fraction 
with  one  non-periodic  element,  which  is  half  the  last  partial 
quotient  of  the  recurring  portion.  This  law  holds  good 
for  every  quadratic  suixl. 

Note  in  the  above  example  that  the  last  partial  quotient 
in  the  recurring  portion  is  an  integer  +  the  given  surd. 

The  following  is  a  very  compact  and  convenient  form 
for  working  such  examples : 

4/8  =  2  +  4/8-2  =  2+  --i , 

i^+2 

4/8  +  2                i/8  -  2       ,    ,  1 
-A =  -^H ~A =  1  + 


4/8+2' 

i/8  +  2  =  4  +  ^/8-2  =  4+  -— ^ , 

4/8  +  2 


412  CONTINUED  FRACTIONS. 


i^  =  i+i:«^^=i  + 


V8  +  3' 
^+3  =  4  +  4^-8  =  4  +  -^^, 

•••   ^=«  +  r+r+r+ !'«*''• 

296.  Theokem  VIII.     An  infinite  periodic  fraction 
may  he  expressed  as  a  quadratic  surd. 

Let  the  partial  quotient  be  1,  2,  3,  1,  2,  3,  etc. 

Then  ^-i     L     1        _1±^ 

±nen  a:-  i  _^  2 -f  3  +  ^~  10  +  Sa:* 

.-.     10a;  +  3a;^  =  7+ 2a:. 

.-.     3a;2  +  8a:  -  7  =  0, 

.-.     a:  =  l/3(  1^  -  4). 

EXERCISE  CXLIX. 

Express  the  following  as  periodic  continued  fractions : 

1.    Vl.  2.    '/13.  3.    V2. 

4.     V6^  6.    Vl7.  6.    4/19. 

Express  the  following  continued  fractions  as  quadratic 
surds : 

111  14.2.111 

"^      2  +  2  +  2+...     *■         +2  +  3  +  2+3  +  ... 

11111 
9-    1  +  2  +  3  +  4+1  +  .. . 


ANSWERS. 


EXERCISE  1. 

1. 

105. 

2.     525. 

3. 

2625. 

4. 

26460. 

5.    85050. 

6. 

396900. 

7. 

91875. 

8.    1701000.  • 

9. 

165375. 

10. 

1181250. 

11.    5a  cts. 

12. 

120a  sq.  in 

13. 

Wmn, 

14.    25a^c. 
EXERCISE  II. 

1. 

67f. 

2.    35i. 

3. 

1130. 

4. 

Same. 

5.    Same. 

6. 

Same. 

EXERCISE  III. 

1.    2aW  +  lOa^^  +  12.  2.  a  -  I2b^  +  3. 

3.    Qx^y  +  5  -  5b\  4.  Sa'y  +  9«/  -  7. 

6.    7a3a;-5aV  +  6. 


1,    :?;  =  12. 

6.      X  =  2S4r^. 


EXERCISE  IV. 

2.    «/  =  7.       s.     z  =  7^. 


a;  = 


4.     X 

c  - 


15. 


1.  x  = 


9a 


EXERCISE  V. 

1.  27  and  36.  2.    45  and  58.  3.    30  and  120. 

4.  17  and  85.  5.    75,  150,  and  225. 

6.  72,  36,  and  12.  7.    525,  175,  and  35. 

8.  Harness  $45  ;  horse  $135  ;  carriage  $270. 

9.  History  $1.38  ;  arithmetic  69  cts. ;  speller  23  cts. 
10.  Sister's  age  10  ;  boy's  13  ;  brother's  18. 


2  ANSWIJRS  TO  QILLETS  ALGEBRA. 

EXERCISE  VI. 

1.    70  and  105.  2.    27  and  45. 

3.  $1.20,  $1.80,  $1.35,  and  10.54. 

4.  $45000,  $30000,  124000,  and  $18000. 

5.  $100.00,  $25.00,  and  $200.00. 

EXERCISE  VII. 

1.  3«  -  4Z>  -  ^ah  +  ^ad  +  6. 

2.  3m  -\-  Aifi  —  20cx  -\-  26cy  —  5c^. 

3.  7  +  24c  -  32^  -  122;.  4.    5x  ~  ab  -  ac  +  la. 

5.  18m  +  16^  -  24^>  +  32c.    6.    2a;-f  62:+21,  or  8a:+21. 

7.  52;-3(«  +  2^»-3c)  +  9;  5a;  + 3(- <*- 2^»  +  3c)+ 9. 

8.  7«^-4c(2^>-4f?-6c)+3;  lah+4.c{-U-^4.d-\-^c)-\-'d. 

9.  27-2«2(- 3c  +  5Z»-6);  27  +  2«2(3c  -  5^  +  6). 
10.  10a;  -  5(  -  42;2  -  5  A  +  7) ;  lOa;  +  5(42;^  +  5  A  -  7). 

EXERCISE    VIII. 

1.    8  and  12.  2.    3  and  9. 

3.  Harness  =  $60 ;  horse  =  $180 ;  carriage  =  $480. 

4.  $2000  the  first  month,  $5920  the  second  month,  and 

$23720  the  third  month. 

6.  20.  6.     36.       7.     18.  8.     132.  9.     99. 
10.    25,  48,  and  46. 

EXERCISE   IX. 

1.  245  bushels  in  all,  98  bushels  of  rye,  and  70  bushels 
of  barley. 

2.  231  in  all,  154  baldwins,  and  42  greenings. 

3.  First  and   second  30   miles,    second  and  third  32 
miles,  and  first  and  fourth  80  miles. 

4.  Louis  had  320,  and  Howard  80. 

5.  First  77,  second  81,  and  third  68. 

6.  Winning  candidate  18156.  Losing  candidates  17344, 
17624,  and  17400,  respectively. 

7.  M  to  N  21  miles,  N  io  6' 6  miles,  and  6^  to  ^81 
miles. 


ANSWEE^S  TO  OILLET'S  ALGEBRA. 


1. 

4. 

7. 
10. 
12. 
14. 
16. 
1&. 
20. 
22. 
23. 
24. 


6. 

5. 

-  6. 

18. 

10. 

-  6. 

0. 

15. 

-  18. 

EXERCISE   X. 

1.     18.  2.     6.  3.     18.  4. 

6.     -  6.        7.      -  18.      8.     -  18.        9. 

.1.     0.  12.     0.  13.     0.  14. 

.6.     6.  17.     '^a.  18.     —  2«. 

.9.     12  +  (+6),  6  +  (+12),  and  12-(-6); 

12  +  (-  6),   -  6  +  (+  12),  and  -  6  -  (-  12); 

a -{-{-a),  a  +  (-a);   -G+(-12), 

-12+ (-6),   -6- (+12). 


EXERCISE 

XI. 

U  B.C. 

3.      40  A.D. 

30  B.C. 

6.      b  B.C. 

C  A.D. 

9.    20°  below  zero. 

11. 

Has  risen  16°. 

13. 

Has  fallen  8°. 

15. 

17°  warmer. 

17. 

12°  warmer. 

19. 

5  miles  south. 

21. 

6  years  older. 

6  B.C.  2. 

a  A.D.  5. 

50  A.D.  8. 

Has  fallen  12°. 
Has  fallen  7°. 
Has  risen  a°. 
8°  colder. 

3  miles  west. 

4  years  younger. 
2  years  younger. 

The  grocer  owes  Hermon  3  dollars. 

7  pounds  less.  25.     20000  dollars  poorer. 


1.    c 


a  —  b, 


EXERCISE  XII. 

-{a  +  b). 


2. 


m 


a  +  b 


3.    In  6  hours.     First  will  have  travelled  24  miles,  and 
second  18  miles. 


4. 


a 


m  -\-  n 
miles. 

6,    50  and  58. 


hours.     First 


7na 
m-{-  n 


miles,  second 


7ia 

7W  + 


ANSWERS  TO   GILLET'S  ALGEBRA. 


EXERCISE 

XIII. 

1. 

25«.                    2.     ^loj^x 

;. 

3.      -  36«52. 

4. 

-  56rc.              6.     -  Qx^. 

6.     2ac^x. 

7, 

2y^  -  2ac  -  5. 

8. 

4.a^x  —  ax^  —  ^ab  —  8. 

9. 

—  dx-{-  Qab  +  c. 

10. 

5a;2  -  aW  -  c  +  7. 

11. 

19/12;?;  =  lj\x. 

12. 

-  V12y. 

13. 

16(«  +  b). 

14. 

-  ft  -  (x  +  y). 

15. 

5(a  +  b)  -5(m  +  ?i). 

16. 

4:a(b-\-x). 

17. 

c{a^  -  h^). 

18. 

-  2az  -  4. 

19. 

0. 

20. 

2a-  b-\-5c-}-3d. 

21. 

4:^  +  3y  +  2  +  5^. 

22. 

c?  —  xy. 

23. 

«2^,3  _p  ^2^^ 

24. 

3ft  +  lOc  -\-^d-x. 

25. 

X -\- b  —  c -\- d. 

26. 

First  7500,  second  7000, 

27. 

(4:X  -  50)  dollars. 

third  6500,  and  fourth 

28. 

8000  dollars. 

6000. 

29. 

{a  +  Z')a;  —  mq 
5 

EXERCISE 

XIV. 

1. 

(«  +  m)x  +  (*  +  ^)2/- 

2. 

{mn  +  pq)x  —  2by, 

3. 

(3  +  6^  +  la)x  -  6y  -\-  m 

H-^. 

4. 

8(a-{-b-\-  l)x  +  (5  - 

■  10)^. 

5. 

x-}-8  and  2a;  +  8. 

6. 

Albert  is  12  and  Howard  24, 

7. 

In  9  hours.     72  miles  and  54  miles. 

2     '       .2 
9.     (ft  —  m)x  -\-  {b  —  7i)y  -\-  {c  —  p)z. 

10.  2(d  -  f)x  ^^e-d)y^  4(/  +  e)z. 

11.  l/12(8ft  +  9%  --  2(1  -  3ft)^. 

12.  (2ft  -  U)x  —  (4ft  +  b)y. 

13.  Herbert  is  l/2ft,  and  Horace  a. 

T         ct      ,  ab        .,  ,      ac        ., 

14.  In  7 hours,   7 miles,  and  ^^ miles, 

J  —  (J  '   b  —  0  b  —  c 


ANSWBB8  TO  GILLET'S  ALGEBRA. 

EXERCISE  XV. 

1.     —  ^x  —  y  +  14:Z.  2.     4«  —  ^  +  2c. 

3.     8a'  -  2a^  +  4«2  _  I5a  +  14. 

4. 

6. 

8. 
10. 
12. 
13. 
15. 
17. 
19. 


21. 


20A2  +  IQa'x.                 5. 

4«3  -  2. 

4/3a:2  -  l/2x  -  1/2.        7. 

«  —  5  +  c. 

-  2a  -  95  -  8.               9. 

X  -  8«2a;2  -f  12. 

2^«  -  7^*  -  3.                   11. 

9  and  18. 

A  has  172.50,  and  B  $77.50, 

-  X  +  tj.                             14. 

2x  -  11. 

5:^:  +  4^  +  7«.  -  11.       16. 

a^  +  ^  +  ^  4.  1, 

4(«  +  h).                             18. 

2a{c  -  x). 

2«2(J_a;)+4«J(«-5).      20. 

x-8. 

6fl  +  m         ,    bm  —  6fl^ 

6 


EXERCISE   XVI. 

1.     —  Sab  —  m  —  2ax,  2.     ox  —  2a. 

3.     2b  —  4:C.  4.     lOa^  —  '^y  -\-  5^. 

5.     —  9ax  —  2by.  6.    0.        7.  0.         8.     3m. 


1.     X  —  {a  -{■  b). 
3.     a;  —  (—  a  +  3a^  — 
5.     a;  —  ( —  a;  -|-  2«  - 

7.  ic— (— «-|-^— ^— ^ 

9.  x—{—2x-\-2m- 

11.  ic—  (2m  +  3a  — 

13.  x  —  (a  -\-  b  -\-  p 

EXERCISE    XVIII. 

1.  m^p-\-q-\-a  —  b  —  c-\-d. 

2.  m-{-a  —  b-{-p-\-q  —  n-\-k, 

3.  15«a;  —  i:by.  4.  0. 

5.  p  -\-  b  -\-  s  -^  t  -\-  m  -\-  n. 

6.  $8360,  $16120,  $23880,  and  $31640. 


EXERCISE 

XVII. 

2. 

a;  —  (m  +  ^^)' 

%).        4. 

a;  _  (35  -  2c  -  5^). 

-  2^^).      6. 

^_(_3+«  +  5). 

n^7i).    8. 

a;  —  (—  a;  —  rt  +  5). 

-2m).    10. 

:^-—  (—2x  -{-2ab  —m) 

2b).        12. 

x—(2am-{-b-{-p—q—7i) 

-  q  -\-  m 

-»). 

ANSWEES  TO   GILLET'S  ALGEBRA. 


a-\-  b  -\-  c 


3b -\-  c 


and 


a  -\-  b  —  c 


4         -  4  ^    2 

8.     llax.  9.     —  2ax  —  6by 

10.     —  2x  -\-  2y.  11.     —  4:bz. 

12.  -  (m  +  6)^  +  2x  -\-  iab  -  5. 

13.  6,  18,  36,  54,  and  72. 


C2!. 


1. 

4. 

7. 
10. 
13. 
16. 
19. 
22. 
25. 
28. 


1. 
3. 

5. 

7. 

9. 
10. 
12. 
14. 
16. 
17. 


21ab. 

—  a^b^xy^. 
a^bcdm. 
30a%hnx^. 
IQSabkni^x^. 

—  SOagxYz. 
A:bc^gnx?z^. 

—  2ia'^xY' 
a^bx^y^. 


2. 

5. 

8. 
11. 
14. 
17. 
20. 
23. 
26. 
29. 


EXERCISE  XIX. 

3Wb. 

?>a%^x^z^. 
.    —  abcdx^. 

10  ba^7n^xy^. 
,    bg?)}  ny'^. 
.    Iba^bhix^yz. 
.    -  ^alrc'j^y. 

—  '^((^xhi^. 

—  apqx^y'^. 

—  2/bacmhiH^. 


3. 

6. 

9. 
12. 
15. 
18. 
21. 
24. 
27. 
30. 


—  32AY. 

—  4:2mVxY. 

—  d^b'^cxK 
^Labmn. 
4:axY. 

—  4:abgxyz*. 
4:abexy. 

—  m^n^x^. 
Za%cdh^K 
Zo^bcxy'^, 


EXERCISE   XX, 

2.     X  -  40. 


\.     X  =  20. 

3.  $19000,  $9000,  $12000,  and  $7000. 

4.  9,  10,  17,  19,  and  2G. 

EXERCISE  XXIII. 


-  3G.t2//2^  -  \:%xy'^zK  2.    a%^c^  - 
3x^  +  Zxy  +  Zxz.                4.   a^bc  —  ab^c  -\-  ab(^. 
a%^c  -  id)^(^  +  a^bc'.          6.    lia'b^  +  2MbK 
15a;y-18.i;y+24;z;y.      8.    56a:y  +  40a;y. 

-  bx^z^  -j-  Sx^z^  —  Sx^y^z^. 

-  4:Sx^yh^  +  9QxYz'-      U.    91a;y  +  106x^y^ 

-  8xYz^  +  lOxH/z'.  13.      -  «2Z>V  +  a^JV  _^  ^2J3^2, 

a^'^c  -  a%h  +  aWcK       15.    -  W  +  9/^a^  -  Uq, 

-  5/22^2  _^  5/3^_y  _^  10/3^-. 


ANSWERS  TO   GILLET'S  ALGEBRA.  7 

18.    -  2a'x'  +  7/2aV.  19.    5/2^4^2  _  5/3^3^3  _^  ^2^4^ 

20.    21/2xhj  -  xy.  21.    l/2a:y  -  3^y. 

22.    —  x^  -\-  l6/4:9xy. 

EXERCISE  XXIV. 

1.  x^  —  1.  2.    x^y  -f  a^y  +  xy^. 

3.  -  3:^:5 _^  dx^-dx^-{-12x^     ^.    x^-\-x^—x-l. 

5.  2:4  +  a:^  +  2,^2  -  x -{- 3.      6.    x*  -  ISa;^  +  36. 

7.  a:^  —  ?/^.  8.  x^  +  y^  9.    ^'C^  +  x^y^  -\-  y^. 

10.  a;54-5ic4+10x3+10a;2  +  5a:+l.        n.    a;  =  11. 

12.  $20000,  $52000,  and  $48000.         13.    5. 

14.  x^  -  5ax^  +  lO^V  -  13«V  +  ISff^a;  -  Qa\ 

15.  x^  —  4«V  +  Stt^a;  —  a^ 

16.  x^-\-2x^y—xy—4:a^y^—xy-\-2xy^-{-y^.     17.    ^*  —  a'^. 

18.  •'c^  —  (^*  —  d  +  c)a;2  —  (be  —  ca  -{-  ab)x  -\-  abc. 

19.  \  —  X  -^  x^  —  X?  -\-  2x^  —  a^  -\-  x^  -\-  a^, 

20.  a^  —  b^.  21.    27^^  —  64«/^. 

22.  125aV  +  27%«.  23.    64aV  _  i25^9a;3^ 

24.  $300,  $550,  and  $350.       26.    8. 

EXERCISE    XXV. 

1.  9x^  +  dx^  -  2x^  +  62;  -  4. 

2.  a;«  +  ^^  -  2a;«  -  2x^  -  5x^  -  x^  -}-  5x^  +  9. 

3.  2x^  -  lOa:^  +  5x^  -  222;^  -  52;2  -^  5a;  +  1. 

4.  2x^  -  7a^  +6a^  +  da^  -  3x^  -f  4a:  -  4. 

5.  1  -  6x^  +  5x\  6.     1  -  7x^  +  Qx\ 

7.  1  +  :6-  -  8a;2  +  19x^  -  I6x\ 

8.  4  -  9x2  _j_  X2a;3  -  4:f4. 

9.  x^  +  .-c^  -  2x^  -  x^  -\-  x^  +  ^  -f  1. 
10.  2x^  -  bx^  4-  2x^  +  6x^  -  3x\ 

EXERCISE  XXVI. 
1.     5xy.  2.     'Sa\  3.     9«2.  4      7^2^. 

6.  —  17:r.  6.      —  ll.r^^.      7.      5z^.  8.      9«c2. 
9.     2xy.          10.     —  3«2^.      11.     i/ba^y. 

12.     —  9a;2?/2^;3,  13,     15(x  +  y)V. 


AJVSWEBS  TO   GILLETS  ALGEBRA. 


14. 

-  39(«  -  hfxK 

15. 

-  'dOcd(a  +  bfxy. 

16. 

bQa%\c  -  ciyxyK 

17. 

b(a  +  bfx. 

18. 

-  ^ac{h  -dfy. 

19. 

Ud(b  +  c)x. 

20. 

2a^c\ 

21. 

-  2Wy. 

22. 

X  =  60. 

23. 

48  and  132. 

EXERCISE  XXVil. 

1. 

x^  +  xy  +  y\ 

2. 

a^  -  ab  -\-  b. 

3. 

a^  -  ?>a%  +  1)\ 

4. 

8a;3  +  mx^y  +  21y\ 

6. 

6/6a*-l/5a'b-l/da^^ 

.    6. 

-  2a%  -  4:ab\ 

7. 

6x^y  -  6xy^  +  8xy. 

8. 

2a  -  Sb  -\-  46'. 

9. 

dx  —  2y  —  4. 

10. 

2/3«  -  l/6^>  -  c. 

EXERCISE   XXVIII. 

1. 

X  -  S.                 2.     x-\- 

3. 

3.     X  —  7. 

4. 

X  -  2.               5.     2x  - 

-  3. 

6.     Sx  +  8. 

7. 

4:x  -  3.             8.     5x  +  4. 

9.     7x  +  5. 

10. 

x^  +  xy  +  y^. 

11. 

^  +  y- 

12. 

9  A2  4-  12abx  +  16^^ 

13. 

2aV  -  3(^b\ 

14. 

7a:2  +  5xy  +  2f, 

15. 

^3  _  2a;2  -I-  a;  +  1. 

16. 

f  -  3x^  -h  2:?;  -  1. 

17. 

.^2  -  a;y  +  y^. 

18. 

x^  +  X  -  y. 

19. 

x^  -  2x  -h  3. 

20. 

x^  ^  5x-\-  G. 

21. 

7«2  -  SrtZ*  +  2bK 

22. 

8.             23.     -  8. 

24. 

5.              25.    —  5. 

26. 

-  18.      27.     5. 

28. 

-  10.      29.    10. 

30. 

-5.        31.     7a- -45. 

32. 

0.              33.    -392-+2 

84. 

X  +  1/3. 

35. 

x^  -  \/2x  -f  3/4. 

36.  \  —  x-\-^  —  Qi?-\-x'^~  etc. 

37.  \\2x^2x^  ^  27?  +  etc. 

88.     2{x  -  yf  -  4.{x  ~  yf  -  {x  -  y), 

EXERCISE  XXIX. 

1.     X  =  5^..  2.     X  =  —  2.  8.     X  —  3. 

4.     a:  =  20.  5.     a;  =.11.  e.     31  doz. 

7,     8  sheep. 


ANSWEBS  TO  GILLETS  ALGEBRA. 


-  27/64a;2^ 


EXERCISE  XXXII. 

2.     x^if.  3. 

6.      —  125a;y.  6. 

8.     dh^.  9. 

11.     49ai»Z>V.  12. 

14.     -  27a».r^  15. 


-  32a:iy 

4/9a^^io. 


EXERCISE   XXXIII. 


«2  +  ^ab  +  9Z>2. 
a;^  —  Vdxy  ■\-  25«/^. 
9a;^  —  %xy  +  «/^. 
81a:2  -  36a^y  +  42/2 


16 

^2 


4"  ^ahxy  -{-  ^Wy'^. 

-  82;  +  .t2. 

-  '2/Ux  +  1/9^*2. 


a;'^  -\-  lax  -\-  (^ 


2. 

4. 

6. 

8. 
10. 
12. 
14. 
16. 
18. 


«2  -  6a5  +  952. 
A,x^  -f  12  2;?/  +  9?/^. 
^x^  +  30a;?/  +  25f/2. 
"IWW  -  lOabc  +  c2 
x^  —  2abcx  +  c^lt^^, 
x^  -  2.^2  +  1. 
x^  +  4/3«2;  +  4/9«2. 
^2  _  3^a;  +  9/4a'2. 
16  +  8a;  +  a;?. 


EXERCISE  XXXIV. 

1  _j_  4:^:2  _^  9^4  +  4a;  +  6a;2  +  12a;3. 
1  +  4a;  +  10a:2  _|_  00.^3  _|_  253^4  j^  24a;5  +  16a;^ 
\j^^x-\-  10a;2  +  20a;3  +  25a:4  +  34a;5  +  36a:«   +  30a;^ 

4-  40a;8  4-  2e5a:io. 


4.     a^^h^^c^^^^-  lab  +  %ac 


+  40a;«  +  25a;io. 
Uc  -  2ad 

+  Ud  —  2cd. 
9^2  4-  452  ^  ^2  _|_  ^2  _^  i2ab  -  Qac  -  Uc  +  6ad. 

4-  4Z»^  -  2cd. 

EXERCISE  XXXV. 

x^  +  2ax^  4-  3  A  4-  a^.     2.     a;^  —  3«a;2  4-  3  A  —  «^. 
a;^— 6a;2|/4-12a;^2_g^3^       ^    3^3  _|_  i2a;22/  4-  Qxy-^  -{-  y^.  ^ 

27a''5  -  U6x^y  +  226xy^  -  12byK 
a^b^  4-  3a2^,2c  4-  3«5c2  4-  c^ 
8^3^,3  _  36«2^,2^  _|_  54^^,c2  -  27c3. 
125^3  _  75«2^6'  4-  15a^>V  _  js^. 


10 


ANSWEMS  TO  QILLETB  ALGEBRA. 


10. 


Ux^  -  240a;'*3/2  +  300a;y  -  125?/«. 


EXERCISE  XXXVI. 

1.     a^-2x^  +  x+l.  2.     l/2a;2  -  1/3^2/  +  1/4/. 

8.     4  and  9.     4.     5  and  8.     5.     3  and  5.     6.     42. 


EXERCISE  XXXVII. 

o  -4-   ft'>'3^y9 


6.     x^y^z. 


—  x^y^. 


EXERCISE    XXXVIII. 

a^-\-2a-  1.  2. 

2a^  —  3  A  —  fl^a:^.  4. 

•2a4  +  4«V  -  4c4.  6. 

4a;2  -  2^2;  -h  2*2.  g 

a;^  —  2a;2?/  +  '^xy^  —  y^.  10. 

52;^  —  ?tx^y  —  4x^/2  +  y^.  12. 


^^  -  ^«/  +  y^' 

dx^  —  4:Xy^  —  2/. 
2x^  -  5a;  +  3. 
^  -  2x^  +  3a;  -  4. 
2  -  3«  -  a2  -j.  2^3. 

a;2  —  l/2xy  —  y\ 


ic^  —  2xy  -\-  y''^ 


EXERCISE  XXXIX. 

1.     106929.  2.     14356521.  3.  714025. 

4.     25060036.  5.     387420489.         e.  25836889. 

7.     .00092416.         8.     .00000784.  9.  4816.36. 

10.     1867.1041.       11.     1435.6521.        12.  64.128064. 


3789. 
2.1319. 


EXERCISE  XL. 

2.     5006.  3.     5083.  4.     129.63. 

6.     ,9486+.      7.     2.4919+.  8.     .0923+. 

EXERCISE  XL!. 

1.     «3  _^  3^2  _^  3^  _|.  1,  2.     x^  +  Qx^  +  12a;  +  8. 

3.  «V  —  TiaH^y^  +  3aa;/  —  /. 

4.  8m3  -  12m2  +  6m  -  1. 

6.     64^3  -  144^2*  +  108aZ>2  _  27*^^ 


ANSWERS  TO  OILLET'S  ALGEBRA.  ll 

6.  1  +  3a;  H-  Qx^  +  Ta;^  +  Qx^  +  'ix^  +  a:^ 

7.  1  -  C^x  +  21a:2  -  44?;=*  +  Q>^x^  -  54^^  +  27:?;«. 

8.  «^  +  6r?2/;  -  3r^2^  +  12rtS2  _  i^ahc  +  Sr/c^  +  8^^ 

-  I2}pc  +  6^>6'2  -  c^. 

9.  8r^6  -  36«5  -f  66^4  -  G3rr^  +  33^2  _  g^^  _^  ]^ 

10.     1  -  3a:  +  6a;2  -  \0x^  +  12.^'^  -  12a:5  +  lO.c"  -  6a;' 

+  3a;8  -  a;^ 
EXERCISE  XLil. 
1.      1  —  x.  2.      1  H-  2a:.  3.      2a:  —  3^. 

4.     3a;^  —  z^.  5.     a-\-  8b.  6.     4a:  —  3a:2. 

7.     1  +  ^  +  ....       8.     l---_-_--etc. 

EXERCISE  XLIil. 

1.     2460375.  2.     11089567.  3.-  1191016. 

4.  17173512.          5.     109215352.  6.  102503.232. 
7.     820.025856.      8.     20910.518875.  9.  056623104. 

EXERCISE  XLIV. 

1.  478.  2.     3.84.  3.     4.68.  4.     9.36. 

5.  27.55.         6.     1.357  +  .     7.     .5848+.     8.      .2154+. 
9.  1.587+.   10.      .7368  +  .   ll.     3.045+.   12.     2.502  +  . 

13.  9a\x^-\-6aWxy+4.bY.     14.     x  =  2. 

15.  15  ft.  by  12  ft.  16.     2«3  +  4c2. 

17.  X  =  1.  18.     48  ft.  by  40  ft. 

19.  90  of  port  and  150  of  claret.  20.     44. 


EXERCISE 

XLV. 

1. 

14a:2  -  43a:  +  20. 

2. 

20a:2  +  62a:  +  48. 

3. 

28  -  47a:  +  15a:2. 

4. 

30  -  20a:  -  80a;2. 

5. 

x^  +  16a:  +  63. 

6. 

a:2  -  8a:  +  15. 

7. 

x^  +  3a:  -  54. 

8. 

a:2  -  4a:  -  77. 

9. 

x^-x-30.  ■ 

10. 

a;2  +  3x  -  28. 

11. 

x^  +  6a:  +  9. 

12. 

a:2  -  8a:  +  16. 

13. 

a;2  -  64. 

14. 

z'^  -  36. 

12  ANSWERS  TO   OILLET'S  ALGEBRA. 


15. 

d6x^  +  39.i;  -  108. 

16. 

72a;2  +  12a;  -  24. 

17. 

24:X^  -  19x  -  175. 

18. 

18a;2  ~^x-  180. 

19. 

4:ax^-(5a-lr^b)x-\-bb. 

20. 

18aa;2+(24«+6c)a;+8c. 

21. 

ba^x^  +  (^^  —  5«c)a;  — 

be. 

22. 

{2a^  +  2a*)a;2  -  («J  + 

V- 

2ac)a;  —  5c. 

23. 

24  -  36a;  -  108^2, 

24. 

63  -  44a;  -  32a;2. 

EXERCISE 

XLVI. 

1. 

20^:4  -  47^2  _^  21. 

2. 

21a;8^47^4^20. 

3. 

30:?:«  -  IQx^  -  32. 

4. 

42a;io  +  4a;5  _  g^ 

5. 

2^4  _  20^2  _  96^ 

6. 

54^2  -|.  3^6  _  rj^j.^ 

7. 

x-\-  12Vx-\-35. 

8. 

6.a;  —  2  |/^  —  48. 

9. 

a; -49. 

10. 

9a;  H-  24  V^4-  16. 

11. 

c^-5. 

12. 

m  —  5. 

13. 

6m^  -  18m^  +  16. 

14. 

3n^  H-  21?i3  -  180. 

15. 

510  +  55  _  5g^ 

16. 

«i4  _  2aT  _  99, 

17. 

a:8  -  49. 

18. 

m«  -  36. 

19. 

4:X^  -  16. 

20. 

26a^x^  -  9. 

21. 

9x  -  175. 

22. 

36a;  -  147. 

23. 

a;2  +  4. 

24. 

4a;4  4-  45. 

3. 


EXERCISE   XLVII. 

{a  +  a;)2-3(«  +  x)-2%  =  a;^  +  (2«-3)a;+a2_3«_28. 
(m  +  2;)2-f  m+a;— 72  =  a;^— (2m  — l)a;  +  m2+  m  —72. 
\x-bf  ^  ^{x-b)-^h  =  x^-{2b-\)x  -^b^-^b-^:b. 

4.  (a;-m)2-5(a;-m)-84=a;2-(2m-f  5)a;+m2+5m-84. 

5.  a;2  —  m  4-  5.  g,    a;2  _  3  _^  ^^ 

7.  (a;  -  4)2  -  {x  -  ay  =  (2a  -  8)x  -  d^  +  16. 

8.  a;3  +  a;2  +  a;+l.  9.     1  -  3a;2  +  2a;4. 
10.    X  —  5^.          11.    5|  days.  \%,    30  min. 

13.  {x  -  5)2  -  (a;  +  6)2  =  -  22a;  -  11. 

14.  \x  +  7)2  -\x-  5)2  =  24a;  +  24. 

15.  X  -  23.  16.  X  -  23.  17.  a;2  -  a;  +  5. 
18.  11.  19.  3.  20.  220  -  16a;. 
21.  7a;  +  148.         22.    a;  =  -  If.          23.     4i  liours. 


ANSWERS  TO  OILLETS  ALGEBRA.  13 

EXERCISE  XLVIII. 

1.    x^  -h  aK  2.    x^  +  27.  3.    x^  -  343. 

4.  a;3  -  (^.  6.    8a;<'  -  27a^ 

6.      a;2  _|.  4^  _|_  lt5.  7.      4«V  _  ;^4^^2  _|_  49, 

8.  ■  a:«  4-  ^x'  +  3a:4  _^  4^3  _|_  3^,2  _|_  2a;  +  1. 

9_  1  _  9.^2  _^  33^4  _  (.3^0  _^  66^  _  36^10  _^  s.'cil 

10  a^j^\/%lb\  11.    l/8rtV  -  8/27^'V. 

12.  l/125«V2+l/216^>%i^    13.    9«V  +  «V  +  1/9  A2. 

14.  l/16««a'i'^  -  l/Ua^h^x''^  +  1/36Z»V^ 

EXERCISE    XLIX. 

1.  x^  +  8.r  +  16.  2.  m^  -  10m  +  25. 

3,  x^  -'dx  +  9/4.  4.  w2  -  5w  +  25/4. 

6.  x'  +  72;  +  49/4.  6.  if  -  ^y  +  81/4. 

^  ^2  _  3/4a;  _|_  9/04.  ^  ^}  +  5/6^  +  25/144. 

9.  x"  +  Z'.r  +  ^^74.  10.    a;2  -  5Z^:c  +  25^>V4. 

11.  x^-^x-^  1/4.  12.    ^^  —  ?/  +  1/4. 

EXERCISE    L. 

1.  x^  +  6a;3  4-  9.  2.  m^  -  Vlw?  +  36. 

3  ^4  _  5^2  _^  25/4.  4.  «»  + 7^4 +  49/4. 

5.  x''  +  ^>a:3  +  574.  6.  2;*  -  2;2  +  1/4. 

7  ,^10  _  2/3x5  _|_  1/9.  g^  ^^6  _  3/4^3  _|_  9/64^ 

9.     (^+2)H6(^+2)+9.        10.     (:^-5)2-3(x-5)+9/4. 

EXERCISE  LI. 

1     ^2  _  8a;  +  16  -  18.  2.    a;2  -  12a;  +36-6. 

3_    r^i  +  7a:  +  49/4-52/4.      4.    a:2_7^+49/4_233/20. 
5^     1/1 6a:*  +  1/2^;^  +  \x^  +  32a:  +  256. 

6.  27  days.  7.    3^  days. 

8^    ^2  _  9^  +  81/4  -  69/4.    9.    22+ii^+i2i/4_i49/4. 

10.  x^+bx+b'/4:--j^.       11.    y'-bij+b'/4.-'^^. 

12.  16/81a;4  _  4o/27a:3  +  100/9a:2  _  250/3a:  +  625. 

13.  72  miles.  14.    5j\  hours. 


14  ANSWERS  TO  OILLETS  ALGEBRA. 

EXERCISE   Lll. 

\.  x^-{-  '^x  +  9/4.  2.    x}  -  5x  -f  25/4. 

3.  x^-3x+  9/4.  4.     x^  +  9.T  +  81/4. 

6.  {x  +  ay  -  6/d(x  +  rt)  +  25/36. 

6.  x  =  2f .  7.     .^2  +  -.^•  +  by4:a^. 

8.  ?/2  —  n/my  +  7^V4m2.         9.    x^  +  3/2:»2  _|_  9/16. 

10.  z^  -  3z^  +  9/4. 

11.  {z  -  5)4  +  3/7(z  -  5)2+9/196.  12.    a:  =  4:-^. 

EXERCISE    Llll. 

1.  2{x^  +  3/2a;  +  9/16  -f  39/16). 

2.  3(:?;2  _  6a;  +  9  -  13). 

3.  4:(x^  -  'S/2x  +  9/16  +  19/16). 

4.  5(a;2+5a;4-25/4-41/4).    5.    Q{x^-\-7x-\-49/4-^7/12). 

6.  1.4142+.  7.    1.442+. 

8.  7(2;2-9:z;+81/4-53/4).     9.    8(x2-52;H- 25/4-31/4). 
10.  9(.'c2-9:c+81/4-53/4).  n.    10(a;2+7a;H-49/4-81/4). 

12.  n(x^  -  %l\\x  +  1/121  +  32/121). 

14.  m(.3-Vm.  +  — ,--^^^^j. 

EXERCISE    LIV. 

1.  lai^U  +  c).  2.    2«2^(.t2  -  4a;  +  a^). 

3.  5Z»V(J:r  -^(?y  -  !)•  4.     7«(1  -  «2  4.  2a3). 

5.  2a;3(3  +  a;  +  2a;2).  6.     15«2(i  _  15^*2). 

7.  5:^3(^2  _.  2«2  _  3^3)^  g^    19^2(2^3  +  3a). 

9.  (3a:2  —  X  —  \)x.  10.    xif^'lxy  —  Zx -\-  "ly). 

EXERCISE    LV. 

1.  {x^a)(x-  a),  2.    (a:  +  3)(a;-3). 

3.  4(a  +  4)(«  -  4).  4.     \zax  +  5Z')(3aa;  -  5J). 

5.  (9  +  4«a;2)(9  -  ^ax^). 

6.  (7«2a;  +  4«V)(7a2a;  -  4aV). 


ANSWERS  TO  OILLET'S  ALGEBRA.  15 

7.  (x+13){x-l).  8.     0/+5)(^-13). 

9.  {a  +  2)(«  -  6).  10.     {b  +  23)(b  +  1). 

11.  6  ft.  12.     86. 

13.  3(2  +  a){2  -  a).  u.    3«(4«  +  6b){^a  -  6b). 

15.  3«(3rt2  -f  5a:2)(3«2  -  5x^). 

16.  bx{5ax^  +  3.^7/)(5^.T^  —  3:?:?/^). 

17.  (:?:  +  10)(a:  +  4).  18.     {x -j- l){x  -  17). 

19.  (a:+  l)(a:-  11).  20.     (.T  +  29)(^' +  1). 
21.  lOJ^  hours;  21y\  hours.    22.     67. 

EXERCISE    LVI. 

1.  (x-\-b)(x+7).  2.    {x-3){x-9). 

3.  (x-\-i){x-8).  4.     (.T  -  3)(a;  +  10). 

6.  (x-7){x  +  6).  6.     (:^:+5)(.'r-4). 

7.  2(a;  -  8)(^  +  3).  8.    (.tH-5)(3x+ 11). 
9.  (2rc-l)(3a:-7).              lo.     (4^;  +  1)(5.t  +  8). 

11.  (bx  -  3){7x -^  12).  12.    4(7:c  +  5)(2a;-5). 

13.  A  can  do  it  in  17|  days,   B  in  14||  days,  and  C  in 

13^  days. 

14.  2(2  -  x){3  +  4.T).  15.     4(6  -  7x)(2  -  3x). 
16.  (5  4- 3.t)(7  +  4:7:).              n.     {2x  +  7){3x  -  a). 

18.  {ax  —  6){bx  +  7).  19.    {ax  +  b){cx  —  d). 

20.  {x  —  {a  —  b))  (x  +  {a  +  ^')). 

21.  l{a-}-b)x  +  2)({a-b)x-4:). 

22.  3(a;+  6)(a:  -  3).  23.     7{x  -  Q){x  +  5). 

24.  10(a:  -  2){x  +  7).  25.    5^2(3^;  -  2)(5a;  +  3). 

26.  93. 

EXERCISE    LVII. 


0. 


1. 

44. 

2. 

-  10. 

3. 

x^-x. 

4. 

x^-x  +  1. 

5. 

28. 

6. 

C3  +  ^3  _   J3  _  ^3 

7. 

if  —  if  =  0,      ■ 

8. 

.^6    _    y6    —Q^ 

9. 

if  -  ?/"  =  0. 

10. 

-if  +  .v'  =  0. 

LI. 

f  +  y'  =  2y\ 

12. 

y'  +  y'=.2yK 

16  ANSWERS  TO   OILLET'S  ALGEBRA. 

13.  y*-\-y'  =  2f.  14.     -y-+if  =  0, 

15.  y""  -\-y''  =  2^".  16.    2/"  +  ^"  =  ^^Z""- 

EXERCISE    LVIIi. 

1.  It  is.  2.    It  is.  3.    It  is. 

4.  It  is  not.  6.    It  is.  6.    It  is. 

7.  It  is  not.  8.    It  is  not.  9.    It  is. 

10.  It  is.  11.    It  is  not.  12.    It  is  not. 

13.  It  is  not.  14.    It  is  not.  is.    It  is. 

16.  It  is  not.  17.    It  is.  18.    It  is. 
19.  It  is  not.           20.    It  is. 

21.  x^  +  hT"  +  Z>V  +  h^x^  +  ¥x^  +  ¥x  +  h\ 

22.  x'^  —  h'j(?  +  y^^  —  b^x  -\-  h^  with  —  2^^  as  remainder. 

23.  x:'  -\-  bx*  -\-  b^x^  +  b^^'^  -\-  b'^x-\-b^  with  2b^  as  remainder. 

24.  X'  ~  bx^  +  b^x^  -  Z/V  +  b^x^  -  Wx^  +  b^x  -  b~  with 

2^^  as  remainder. 

25.  X?'  -\-  bx^  -\-  h'^x^  +  V^x  +  b^  with  2b^  as  remainder. 

26.  x^  —  bx"  +  Wx^  -  IH^  +  b^x^  -  b^x  +  b\ 

27.  x^  +  bx^  -\-  b^x  +  b\ 

28.  (^  -  bx^  +  b^x^  -  ^V  _^  ^4^  _  j5^ 

EXERCISE  LIX. 

1.  2:'^  +  a:^^  +  x^y^  -\-  xy^  -\-  y^. 

2  x"^  -\-  x^y  -\-  x'y'^  +  x!^y^  +  a;^«/*  +  x^y^  +  a;;^^  -f  2/'^- 

3.  X?  —  x^y  +  x^y'^  —  2:y  +  ^^'^  —  y^' 

4.  2;^  —  x^y  +  .^**?/^  —  0:^2/^  +  ^2/*  ~  ^^'^  +  y^' 

5.  a;2  4-  3a:  +  9.  6.    x^  +  3a:2+  9^ _|_  27. 

7.  a;3  -  2a;2  +  4a;  -  8.  8.    a;^  -  2a;3_j_4^_8^_|_  16^ 

9.  0.  10.     V'-aK  11.     0. 

12.  0.  13.      0.  14.      -  12^2^. 

16.  W  -  '^V'c  -  Ui^  +  1c\  16.     {x  -\-  2)(3a;  +  4). 

17.  (.T-l)(.T+5).  18.     (a:  +  4)(a;  -  3). 
19.  (.T  -  2)(3a:  -  2).  20.      \x  -  l)(4.r  -  3). 


ANSWERS  TO   GILLETS  ALGEBRA.  IT 

EXERCISE  LX. 

1.    13i  hours.  2.    120  hours. 

3.  50  days,  214  <iays.  4.    36.  5.    27  days. 

6.  A's  $1650,  B's  $1320. 

EXERCISE   LXI. 

1.    5a;^y.  2.    Ix^yh.  z.    ISadcd. 

4!    xy.  5.    I'^a^bh^.  6.    Ux'^  +  Y^^-^ 

^       Qx'^  +  n,^  13^2n  _  l^^n  +  PyQ _  g^m  +  2 _  24^h  +  2  _  202;!'  +  ^ 

8.  Sx""  +  Qx""  -  4:xK 

EXERCISE   LXII. 

1.  cc  +  1.  2.  ^  +  3.                   3.  X  —  10. 

A.  X  —  2.  5.  X  -^  a.                 Q.  X  —  y. 

7.  cc  —  1.  8.  :?;  +  ^.  9.  6(rc  +  l)^. 
10.  x^—y^.  11.  (a;+a)(a;— 2a).  12.  ^  —  y. 

EXERCISE  LXIII. 

\.  x-\-\.  2.    X—  d.  3.    a;  —  2. 

4.  X  —  2.  5.    2a;(.T  —  3).  6.    3A(x  —  a). 
7.  {x  —  af,             8.    5a:2  —  1.  9.     [l—xy. 

10.  a;2  +  4a;+5.     11.    a:2  +  2i»  +  3.      12.    x^ -}- 2x  +  3, 

13.  a:2  -  3a:  +  2.  14.    x^ -\- 2x  -  3. 

16.  18a:"*+3-24a:"*+^+30a:'*+*+21a:2'«+2_28;x;2'»+35a:'«+'*+3^ 

16.  4a;«+''  -  12a;3«+^  +  20a;^  -  6a;2«-2  -f  18a:^-2  -  30a;«+i. 

EXERCISE   LXIV. 

1.  36  min.  2.    8  o'clock. 

3.  5y\  min.  past  10.  4.    5y\  minutes  before  2. 

6.  5y\  minutes  after  4  and  38  \  minutes  after  4. 

6.  26.  7.     69.  8.    16|  hours. 

9.  42  hours.  10.    16 5  years.  11.    35  years. 
12.  40  and  10. 


18  ANSWERS  TO   GILLET'S  ALGEBRA. 


EXERCISE 

LXV. 

1. 

3. 

5. 
7. 
9. 

x^y  —  y^.                            4. 

(X^  -  1)(4:X^  -   1).                       8. 

{x-l)(x-2){x-d),    10. 

60xYz\ 

x^  -  5x^  -  9a:  +  45. 
^3  _^  7^2  _  28a;  -  160. 
n(x  -  2){x^  -  9). 
3(x  -  dafix^  -  4^2). 

EXERCISE   LXVI. 

1.  12x^  +  2aa^  -  \aH^  -  Tia^'x  -  \^a\ 

2.  (4«.  -  ^)(«-  ^)(3«2  + J2). 

3.  (a:2  -  2a:  +  3)(6:6-3  +  a'^  -  44x  +  21). 

4.  {x^  +  5x  +  7')(7a;4  -  40r^  +  TSa:^  -  40a:  +  7). 

5.  a:(a:+l)(:r  +  2)(a:- 2)(a;  +  3). 

6.  x{x-  l)(a:+2)(a:  +  ^){x^  -  2a:  +  4). 

7.  2rt(2«  -  h){2a  -  U){2a  +  3^). 

8.  6a:(.T+  l)(a:  -  3)(a' -  4). 

9.  (3a:  +  2)(8a;^  +  27)(8a:S  _  27). 

10.     3 (a:  -  ZyYix"  -  4/).       n.     a-^*"  -  a:2«. 

12.     9«2«a:2'"  —  IG^^m^^n^       ^g     a:-2*"+i.        ^^     2«™-"a:-2^+2''. 

EXERCISE  LXVII. 

1.  5j^3  minutes  past  11  o'clock. 

2.  In  12  hours.     120  miles. 

3.  12  miles  an  hour.     240  miles.  4.    14f  days. 
5.  33-^  days.                             6.     24  days. 

7.     120  days.  8.     a:^'"  —  x'Hf  +  «/2«. 

9.       X"^  +  a:*"*?/**  +  a;3*»y2i«  _^  ^2m^3»  _^  ^m^4«  _^  ^5»^ 

EXE5^CISE  LXVill. 

«       4^        5c  2a:  hy         Sac  9 


3«  —  5^  +  4  a:  +  ^ 


2«  +  J  •  4c 

7fl;+9^+^--^-18  _  9a: +  66? +  11^  4- 5c 


ANSWERS  TO   GILLETS  ALGEBRA. 


19 


EXERCISE  LXIX. 


13. 

14. 
15. 

IT. 


2a 

— .  2. 

X 

3a; 

— •  6. 

2x-  4, 


x-3 


x^  -  3.g  +  9 
X  —  '6 


Sab 

~W' 
10. 

13. 


3. 

7. 

3^+J. 

a;  +  2' 
3a:  —  y 
4cX  —  y 


1 

a' 

x-{-6 

r 


X 


11. 


14. 


4. 

X  —  a 

X 

x-\-h 

o. 

a; +  3' 

x^-{-4x-^U 

.T+7       ■ 

2x 

-  3 

12«3^3 

~Vlx^  +  a;  -  6  ' 
15«V 


EXERCISE    LXX. 


20«V 


3. 


2a; +  3' 


a;2-9a;  +  18 


x^ 


X 

20a:^  -  53a;  +  35 

8a:2- 
6^V 


42 


34a;  +  30 
-  14a^>V 


21a;2  -  11a; 


40 


3  -  ^iCiWx^     ' 
15a;2  -  9a;  +  42 


7a;  4- 8 
15a;2  +  13a; 


9-  hx 

9a;  +  20 


11 


loa^x 


a;2-  16 
Their  sum  = 
30 


and 


9a^x 
2  +  10;r  +  24 


6  -.  3a; 
and 


r^'a; 


9a=^^a;' 


x^ 


16 


x^ 


x 


2a;2  +  a;  +  44. 
X?  -  16.      • 


14a;  +  48 


x" 


6a 

4rt^ 


a;  -  30    '     a^  -  a;  -  30 

_  a;2  +  25x-  -  43 
~  Tc^"-  a;  -  30  • 
oa;  -|-  6       6rt  —  oa;  —  6 


30 


4«:^ 

45x^+20a;2-3a;+18 
15.t2  +  14a;  -  8     * 
25^>2  -  84rt6' 
36^2 


4^2 


16.      — 


b'^  —  4ac 


18. 


4«2 
4a^>     * 


20  ANSWERS  TO   GILLET'S  ALGEBRA. 

19        iC^*"   ^2rn-\-n  _r_  ^TO-f-2n   ^n^ 

20.     a;^"*  +  aj^"""^"  +  a^^'^+^n  _|_  ^m+sn  _|_  ^«^ 
EXERCISE  LXXI. 
1.     1.  2.     --^— .  3.     1/x,  4.     — ^       -^^ 


5. 


a  X 

X?  -  13a;  +  42  15.^:2  -  26a;  +  8 


a: +  7 


10. 


a'^  —  a;'^ 
2a;-  1 


EXERCISE  LXXII. 

ab 
2. 


7/12. 

«-  11 

a~%    * 

a;  +  l 

a;  +  5*  .   • 

14  -  17a:  - 

-  6a;2 

2«  - 

-  1* 

2a;- 

-  1 

2a;- 

-  3  * 

6a;2 

-  23a;  +  20 

3a;-  2 

2a;2. 

-  21a;  +  27 

6. 


4a;  +  6         '  '  a;  —  5 

EXERCISE  LXXIII. 

1.  x^  -X-  12,    a;2  -  15a;  +  56,    ^x  +  48. 

2.  9a;2  -  9a;  -  28,    lOa;^  —  43a;  -f  28,    5a;2  -f  51a;  -  44. 

3.  -40a;2  +  94a;-48,  -35a;^-19a;-42,  -3(a;2-6a;  + 9). 

4.  a;2  -  64,    a;^  -  64,    7a;2  +  28a;  -  224. 

5.  x'^x-  42,    a;^  -f  216,    5a;2  -  30a;  +  180. 

6.  i/'^"'  -4-  o;S»»+'»  _L_  '7;*"'  +  ^"  -4-  -|.3'«  +  3n  _j_  ™2»i+4n  _j_  ^m+5»  _j_  ^6»» 

EXERCISE  LXXIV. 

1.  A  is  48  and  B  is  12. 

2.  49^  minutes  after  3  o'clock.  a.     17  and  28. 

4.     35  dimes,  5  cents,  and  10  dollars.     5.     98  and  215, 

. ,  cmim  —  1)       _,,  cin  —  1) 

6.     A's  age  =  — ^^ '- ;     B's  age  =  -^ . 

m  —  n  ®  m  —  n 


AI^SWEBS  TO   GILLpT'S  ALGEBRA.  21 

100(^  -  a)  11 

7.  •  8.       TT"* 

ac  3 

9.     27y\  minutes  after  5  o'clock. 

n  —  r         ,    nq  4-  r 

10.     ;— r    and      ^  ,'      . 

S'  +  l  ^  +  1 

EXERCISE    LXXV. 

1.      -3f  2.      11.  3.      7.  4.      -8. 

6.  1.  6.      -  If.  7.      -  10.       8.      -  1. 
9.      7.                    10.      3.               11.      1.  12.      —  2. 

13.      1/4.  14.      -3^-     15.     If. 

EXERCISE    LXXVI. 

1.     5^.  2.     6.  3.     9-1^.  4.     1.  6.     1*. 

EXERCISE    LXXVII. 

1.      1/2.  2.     4f.  3.     4i. 

4.      -4/7.  5.     Hf.  6.     3. 

7.  3i.  8.     -4i.  9.     2^. 
.                                 ab  —  cd 

10.         0.  11.         -^^ j.         12.         2yV«. 

EXERCISE  LXXVIII. 

1.     55  minutes.  2.    37^  min.  and  25  min. 

3.  130000.  4.    $84000. 

6.     A  39  miles  and  B  27  miles.  6.    283.  7.    536. 

8.  Of  the  first  -^^ ^,  and  of  the  second  -^^ z-^. 

a  —  0  a  —  i 

9.  $750  and  $500.  10.    192  miles. 

11.  $15.36  and  $4.56.  12.     Hound  72  and  fox  108. 
13,     300.                              14.    Man  84  cents,  boy  42  cents, 

15.  85  gallons  of  spirits  and  35  of  water. 

16.  1500.  17.    28.  18.    3  shillings. 

EXERCISE  LXXIX. 

1.     Vn^.  2.     V^i:^  3.     V^^. 

4.  V'2ba%\  6.     V^aF.  6.     Vl^\ 


ANSWEBS  TO   GILLET'S  ALGEBRA. 


8.     yl/9«y   or    — d-. 


VU  V9 


>.     V  -77^2-  or  —=.—.     10.     VflJ^  +  2«^>  +  b\ 


11. 

Va;2  -  2xy  +  2/2.              12.     V^a""  +  426^^  _^  49^ 

13. 

fe                                   14.     '^STaV. 

„                                                l//Y9iy3 

15. 

hi 

16. 

\/x^^lbx'^lbx^l2b.     17.     \^a^  -  9^2  _^  27^  _  27. 

18. 

.727«V         |/27«V 
^  ^^0^    "     f  64.3  • 

EXERCISE  LXXX. 

1. 

24/3.                   2.     5  1^3.                   3.     64/5: 

4. 

7  1/15.                 6.     16  V2.                 6.     9  V7. 

7. 

3  1^5".                   8.     4  V'y.                   9.     8  1^11. 

10. 

4.aV¥l).             11.     5«:c2  4/5«.          12.     la^x^V^ax. 

13. 

2a{a  +  J)  1^6?.                   14.     2x^y{x  -  y)  V^xy, 

EXERCISE    LXXXI. 

1.     i/99.                   2.     1^208.                   3.     V252. 

4.     ^^72.                   5.     >^^320.                   6.     ^864. 

7.     VSia^  -  9«2j.                  8.     V^x^  +  Qx^y  +  Zxy^, 

EXERCISE   LXXXII. 

1.     1/2  V2.  2.     1/5  i/5.  3.     1/3  VQ, 

4.     1/6  l/fK  5.     i-  f^2l^.  6.  .  — ^  V¥-b\ 

'  Ix         ^  a  —  h 

1  ./^^-TT.— 7-7^      .  1 


7.      -t-t;  ^^^  +  10:c  +  24.     8.     — ri^  i^^^  +  2a;  -~  35. 


ANSW£JMS  TO  GILLET'S  ALGEBEA,  23 

9.     ^r-V-r  VlOx^  +  x-2.    10.     5-^^  4/12^21146^+42. 
2x  -{-1  dx  —  i 


11.     — ^  ^20  +  7x-Qx\   12.     T y  12a;2  +  7:c  -  12. 

OX  ~j~  4:  'lit/        O 

EXERCISE    LXXXIII. 

1.     18^2".  2.     37  V2.  3.     ^^15". 

4.     2/5  V'e".  6.     25a2a;  4/3^^         6.     -^  V^. 

w(?i  +  Vns) 


7.     ISft^*  V'2«2^2,  8^ 

n  —  s 


EXERCISE    LXXXIV. 

1.     4  V'5.  2.     -  3«2^  f  ^.  3.     2^>  Vb. 

31       3_  

4.     UV2a,  5.     ^VQ.  6.     -l9aVab. 

y  u 


7.     (13c  -  d5cd)  V2i.      8.      (6-  -  X  -~-^  V^^ 
EXERCISE  LXXXV. 
1.      96  Vd.  2.     1^  H.  3.     24  y^. 

4,     1/2  4/6".  6.     ~4/^.  6.     4^% 

7.  64/10  +  7    VT5  +  84/6  +  24. 

8.  6  +  V16.  9.      6  4/2r-  46.       10.     2  l^e. 
11.     6ff  -  6a;  +  5  Vox,  12.     3  4/7  -  47. 

13.  64/5  +  14.  14.  53-144/5. 

15^  32-104/7.      .  16.  a; +  7  4/^+ 13. 

17.  a; +  18  4/^+ 81.  18.  x  +  Vx-dO, 

19.  a; -2  4/3^+3.  20.  -3. 


0^2. 


24  AlfSWEIiS  TO   GILLErS  ALGEBRA. 

21.     15  +  4^11:  22.      V^^  -  3x  -  40. 

23.     2a;  +  2  +  2  i/a;2  -f '2^  -  24.         24.     V^f^^^. 
"•^      2a;  -f  2  |/a;2  -  9.  26.     ISa;^  |/(:j''^  -  13«  +  42. 

24a%  +  8«  t^6^+  4^.  28.     35«Z>a;  +  245«&. 

25a;  -  58  -  24  Vx^  -  x  -  42.       30.     63^3  V^~rT{j. 

34«3  _  98^2  +  30«2  4/«2  _  2«  -  15.      32.     x  -  29. 
33.     x  +  2.  34.     -  13.  35.     -  7a;  -  26. 

36.     9«V  _  72«V  _  25a;5  -  175a;^ 


27. 
29. 
31. 


EXERCISE  LXXXVI. 

1. 

113. 

2.    -  166.                3.    172. 

4. 

-  6. 

5.    a  —  4^.               6.    9c2  —  4a;. 

7. 

X. 

8.    2j»  —  q,               9.    2a;. 

10. 

25a;2+75?/2_49«2.     n.    -  2ax.              12.    2x^ -\- 6x. 

EXERCISE   LXXXVII. 

1. 

44. 

2.    59. 

3. 

«2  +  J2  +  ^2  _ 

-  2«J  -  2ac  -  2hc.        4.    64. 
EXERCISE  LXXXVIII. 

1. 

-  64/3  +  8  V 

104-134/42"             64/7-1/2" 
^-             26          •      ^-            25        • 

4. 

2.                  5.    ^  -  i^a^  -  l^. 

3  _  i/9  _  «4 

6. 

7a;  +  3  +  8  Vi 
3a;  +  15 

8. 

1                              1  +  a;2 

11. 

Vs. 

aVx. 

12.     '^'S.                         13.     44/5. 

14. 

15.    Vx  -  7.             16.    f  :z^^  +  2a;  +  4. 

17. 

V^x-^3. 
Vx  -  9. 

18.    \^x  -  3.             19.    Va;  -  7. 

20. 

21.    4/3a;-2.            22.    Vbx-1. 

23. 

11  -  3  Vf 

3 

3  4/f  -  2  1^3"            19  -  6  i^ 
24.                3            .    25.             ^^         . 

ANSWBBS  TO  GILLErS  . 

AIGEBMA. 

26. 

2+^6. 

^/X'U 
27.     ^. 

y 

».  f. 

29. 

a  —  X 

:    30.  4  +  4/15; 

EXERCISE   LXXXIX. 

1. 

VlOO, 

^^125,      and       ^^11/2. 
'i^(a  -  by,      and 

2. 

'\/{a  +  hy, 

y'Ca^  f  x^y. 

4.    1/2  |/^. 

3. 

-♦/10125. 

6.      t^a^ 

6. 

7.    3/2^8/3. 
EXERCISE  XC. 

1. 

14.          2. 

8.                      3.    20. 

4.   2|. 

5. 

13.          6. 

6/5.                   7.    144. 

8.    2. 

9. 
13. 

4|^.        10. 

ItV                 11-    5. 
14      (^-*)' 

12.    12. 

15.    1/6. 

16. 
19. 

3«/4. 

23.          20. 

17.  (v;^-i)2. 

a  -  1.              21.    42i. 
EXERCISE  XCI. 

18.    2/5. 
22.    9«/10. 

1. 
4. 

24/17. 

12^3_ 

2.     Hh 

a{c  -  1)2 

^-           4.       • 

3.    3|. 

6.    8/45. 

7. 

25/168. 

(«-&)2 

25 


EXERCISE  XCII. 

1.    16.  2.    1/64.  3     1/5.  4.    1/6. 

6.    1/1000.  6.    36.  7.    a^VK 

8.    a-^/'^W^'^.  -  9.    «V3Z,i2.  10.    a-%-y'^. 

11^   «i/5  _|_  ji/2  ^  ^4/3^  j2^    ^^3/2  _|_  ^1/3^2^ 

13.   a^/V/s  +  a^/^^^,  14.   a;2/32/0V3  4-  ^s/v/s^ 


26  AJ)f^8WEBS  TO  GILLETS  ALGEBkA. 


3-1  1 

16.  VX^ 3-.  16.    T"vf* 

18.  — T-  X      S--  +  -r— .  19.    tC^/^  —  ^*/^ 

20.  1  -  ^'/^  21.    «'/'  +  2'^/^  22.    X^-1, 
23.    :r3  +  2a:V2  -|-  3  +  2:?;-^/2  +  a;-3. 

4n  8m    2n  4h 

25.    ic5/2  +  //2.  26.    i«^^  +  ^V  +  2/'- 

27.  a:«/3  _  ^4/3^4/3  _^  ^8/3^ 

28.  iiJ^/^  -  22;«/5i/V4  4-  4a;V5^V2  _  SrrVy/*  +  \^. 

29.  ic2/3-^-2/3. 

30.  «4/10  _|_  ^3/10^1/5  _|_  ^2/10^2/5  _|_  ^1/10^3/5  _^  ^.4/5^ 

31.  ^  +  2/- 

EXERCISE  XCIII. 

1.  x  =  2,  y  =  ^.      2.  X  =  ^,  y  =  b.     3.    a;  =  2,  2/  =  1. 

4.  a:=:4,  i/  =  — 1.    5.  ic  =  1,  «/ =  2.      6.     ^=  — 3,  ?/=:4. 

7.  X  =  6^  y  —  —  Q,  8.    a^  =  —  1,  «/  =  —  2. 

9.  2;  =  3,  _^  =  —  1.  \Q^   X  —  1 ,  y  =  5. 

11.  a;  —  3,  ?/  =  8.  12.    x  —  2,  y  —  3. 

'T*        '7*^         or 

13.  1  -  2  ~  "8~  ~  16"'  ^*'     ''^^  -2^+1. 

15.  a;  =  15,  2/  =  16.  16.    x  =  ^,  y  =  2. 

17.  »:  =  2,  y  =  -  1/2.  18.     x  =  \,y  =  l, 

he  ac 

19.  X  —  b,  y  =  b  20.    .T  =  — r-7'  ^  =  — ^^^^t.* 

^  a-\-  h  ^       a-^h 

h  —  c  a  —  c  _,  07 

21.  X  —  7 ,  y  = ,.    22.    X  =  2h  —  a,  y  =  2a  —  u. 

b  —  a  ^       a  —  0  ^ 

ac  he 

23.    x:=:a,  y  =  h,  24.    X  =  -2x12'  y 


«24_^2'   ^-«2_^ 


AirSWEnS  TO  GILLET'S  ALOEBBA.  2? 

7«  +  8Z/        8«  -h  n 

EXERCISE  XCIV. 

1.  7andl.  2.    8/15.  3.    45.  4.    54. 

5.  58  years  and  18  years.  6.  Each  would  do  it  in  50  days. 

7.  Tea  28  cents  a  pound,  and  sugar  3  cents. 

8.  4  gals,  from  the  first  and  3  gals,  from  the  second. 

9.  2  gals,  from  the  first  and  10  gals,  from  the  second. 
10.  Tea  30  cents  a  pound,  and  sugar  3^  cents. 


EXERCISE  XCV. 

1. 

x  =  2, 

2. 

x=\, 

8. 

a;  =  3. 

^  =  3, 

2/=-2, 

2/ =  5, 

z  =4. 

^  =  3. 

0  =  -3 

4. 

x  =  l, 

5. 

a;  =  4, 

6. 

a;  =  3/2, 

2/ =  3, 

2/ =-3, 

2/  =  2/3, 

z=  -b. 

^=2. 

;2?  =  5/6. 

7. 

x  =  15, 

8. 

i.  =  3. 

9. 

a;  =  9, 

y  =  lS, 

2/ =  6, 

^=18, 

z  =  20. 

z  =  9. 

^  =  6. 

10. 

X  =  S,  y  =  Q,  z  ^=- 

5. 

1. 

5m                 2m 

3a;'  +  4a;'  - 

7n 

-  V^x^^ 

3n                    « 

-  6a:*  +  lO/. 

3»  +  »i 

n  +  3»n 

12.  a;2"  +  a;     '     +»:"  +  '"  + a;    '     +  ic^"*. 

13.  4a:2/3  _^  25a;V3  _|-  iGa:^  -  I2:r  -  24a;S/3. 

EXERCISE   XCVI. 
1.    9,  11,  and  18.  2.    37,  25,  and  16. 

3.    124,  $32,  and  116.   4.    A,  1420;  B,  $640;  C,  $1040. 
6.    A  in  40  days,  B  in  120  days,  and  C  in  60  days. 

6.  A  in  10  days,  B  in  15  days,  and  C  in  12  days. 

7.  234.  8.     253.  9.     428. 

10.    A,  _  ,     ■       .;    B,—— —-———;    C, 


rs  -\-st—rt  —7's  -\~st^rt  rs—st-{-rt 


28  ANSWERS  TO  OILLET'8  ALGEBRA. 

11.  Rate  of  stream,  2  miles  per  hour;  rate  rowing  in 
still  water,  10  miles  per  hour. 

12.  Rate  of  the  current,  3  miles  per  hour;  rate  of  crew 
in  still  water,  12  miles  per  hour. 

13.  Rates  36  and  27  miles  per  hour  respectively,  and 
distance  75 G  miles. 

14.  Rates  25  and  30  miles  per  hour  respectively,  and 
distance  330  miles. 

15.  15  persons,  and  5  dollars  a  piece. 

16.  Number  of    persons   ^,         ^       ;    each    received 

bm  —  an 

om  —  an 

EXERCISE  XCVII. 

1.  {x-\-V^^){x- V^^).2.    {x  ^  Vf){x  -  Vl). 
3.    (^+4  V~i)(x-4:  V^).  4.    3(^  +  V^)(x  -  1/3). 

6.  b{x  +  VE){x  -  Vb).  6.    7(a;4-  V'^)(x-  V^^). 

7.  2{x  +  1/2  Vq){x  -  1/2  |/6). 

8.  3(ic  +  1/3  V-  lb)(x  -  1/3  V-  15). 

9.  5(a;  +  l/5|/l0)(:r- 1/5  VIO). 

10.    ^{x  +  1/4  V-  U){x  -  1/4  ^"-^nj) 

=  4(2:  +  1/2  V'^^)(x  -  1/2  \r^), 

11.  ^x  +  2/3  \^){:x  -  2/3  VI). 

12.    7(a:  +  1/7  V-  'db){x  -  1/7  V^^^), 
EXERCISE  XCVIII. 

1.  ^[x^——-)[x^ — —y 

2.  2{x^\/2){x+^ 

3.  5(.+  IlA4_^)(,  .-^-^^ 


•^•+  10  / 

3  +  2|/ir\/     .    _3-24^> 


/         _3  +  2|/ll\/     , 


AWSWEBS  TO  OILLET'S  ALGEBRA. 


29 


6. 

8. 
10. 
11. 
12. 

13. 
14. 


1. 

4. 

7. 
10. 
13. 


^8> 


J     ,    -4  + l/88\/     ,    -4-  V88\ 


6  a; + 


4  b: + 


3  +  i/- 


^%  +  ! 


-  V-  87> 


i^2> 


8 


2{x^4.){x-\-l).         7.    661  at  125,  and  108i  at  140. 

7(a;  +  l)(a:H-2/7).     9.    1l{x-\-'^  ^  Vb){x^2-- Vb). 

d{x^^){x-  2/3). 

4(a:  -  3  +  t'+'6)(a;  -  3  -  i^+~6). 

15(a:-  3/5)(a:  +  2/3). 

S  +  ^/yN/         6-Vf 


s  —  bm      ,  ^  —  «^» 
and 


■)• 


16.     X 


a  —  h   "        b  —  a 
a{c  -  If 
4c       • 


acres. 


16.    a:  =  4/9. 


-  3,  6. 

5,7. 
3,  -3. 

—  a,  b. 

2/5,  -  4/3. 


EXERCISE  XCIX. 

2.    5,  -  9. 

5.    -  4,  -  4. 

8.    a,  —  «. 
11.    -  3/2,  1. 
14.    -  8/7,  -  1. 


3.  -  5,  -  8, 

6.  5,  5. 

9.  —  a,  —  b. 

12.  4/3,  -  3. 

15.  3/2,  4. 


16.    5/3,  4. 


1.    x^  -  lOx  +  21 


EXERCISE  C. 

0. 


3.  a:2  +  8a;  +  7  =  0. 

5.  x^  4-  9a:  =  0. 

7.  x^  +  16a:  +  64  =  0. 

9.  4a:2  -  15x  +  9  =  0. 

11.  16a:2-  28a: +.11  =  0. 

13.  18a;2+27a:+  10  =  0. 

15.  5a;2  -  33a:  -  14  =  0. 


4. 
6. 
8. 
10. 
12. 
14. 
16. 


a;2  +  2a:  -  24  =  0. 
0:2  —  2a;  =  q^ 
a;2  -  49  =  0. 
x^  -  22a: +121  =  0. 
18a:2  _  18a:  +  1  =  0. 

^.2 


2a:2 


8a:  +  22  =  0. 
-  a:  -  3  =  0. 
6a:  +  4  =  0. 


so  ANSWERS  TO  GILLET'S  ALGBBM. 

17.    x^  -  ix  —  ^  =  0.  18.    x^  —  lOo;  4-  22  =  0. 

19.    x^  -  I82;  +  85  =  0.         20.    25x^  -  35.T  +  13  =  0. 


IP-  —  4«c 


21.    242;2  -  44a;  +  21  =  0.     22. 

a;^-2a; 
^^'      a;  +  2  •  ^^-     2  -  a;' 

EXERCISE  CI. 

1.  1,  -  1/3.                  2.    2,  -  3.  3.  2,  3. 

4.  4,1/4.                         5.    -1,2.  6.  -3/4,-9/4. 

7.  5,  -  6yV                   8.    1,  -  7/32.  9.  a,  1/a. 
10.    3,  13/11.                  11.    2,  1/2.  12.  1/2,  -  3. 
13.    5,  -  1/6. 

EXERCISE  CM. 

2.  5/7,  3/4.                   3.     -  a,  b.  4.     -  3/4,  -2. 

5.  2/3,  -  5/4.             6.     ±  6,  ±  9.  7.  ±  6,  ±  10. 

8.  ±2/3,  ±3/4.  9.     ±^,±^:^. 

EXERCISE  cm. 

1.  15  and  8,  or  -  23/2  and  -  37/2. 

2.  3,  4,  and  5,  or  —  1,  0,  and  1. 

3.  20  and  8,  or  —  14  and  —  26. 

4.  5,  6,  and  7,  or  —  1,  0,  and  1.  6.    4  and  2. 

6.  1,  2,  3,  4;  or  5,  6,  7,  8. 

7.  3,  4,  5,  6,  or  -  4/3,  -  1/3,  2/3,  5/3. 

8.  20  barrels;  6  dollars  a  barrel.  9.    $80  or  $20. 

10.  $60.       11.    8  miles  an  hour.         12.    7  miles  an  hour. 

EXERCISE    CIV. 

1.     -2,  -4.  2.    20,  "6.  3.    5,  -5/2. 

4.     1,  4|.  5.    1,  2f  6.    3,  1/2. 

7.     4,  -4^.  8.     1,  -3/4.         9.     2,  -2/9. 

^        ,  /o                  -6±  VS  ,77 

10.    7,-1/3.        11. .      12.     —a-\-I),—a  —  I?. 


AJSfSWEBS  TO   OILLErS  ALGEBRA.  31 


13.     —a,  —3ak  14. ,  ab,  is.    a,  b. 

(t 

16.    0,  — — Y.  17.    2ft— 6,  db—2a.  18.    «,  1/a. 

a  -\-  b  ' 

19.    ^',  y.  20.    |(-3±^).    21.     ±  V^H^^, 

22.     1/8  (-25  ±4/33).    23.     3/5,  -2/3.        24.     3,  + 1/6. 

EXERCISE    CV. 

1.    30  and  40  miles  per  hour.  2.    40  and  45  miles  per  hour. 
3.     2 1  hours.  4.     2^^^  hours. 

2,,,  1  2|/ft3 

5.    a'^^/^,  6.    — i—  +  ~^^- 

a"  Vb^        Vb^ 

1        '  ^        {m  —  n)bd  ~  cb-\-  dx 


C2  -  ^>2' 


10.    (^  -  2?/)(7a;  -  11). 


EXERCISE   CVI. 

1.    5/2,  3/2.  2.     ±  2/3  1/3,  ±  4/5".  3.     6,  5f. 

4.  ±V^,  ±l/2f6.    6.     5,6.    6.     ±2/3  1^,  ±1/3  t/2r 
7.     ±  1/2  V^,  ±  1/3  |/6.         8.     ±  1/6  V6,  ±  1/3  V2, 

9.     3/2,  -  2.  10.     3/5,  -  4/7. 

11 

13.   -  1,  - 1/2.  14.    t^m,  4. 

EXERCISE    evil. 

1.  3  miles  an  hour,  3^^  hours. 

2.  5  miles  an  hour,  5|  hours. 

db^  -  75ftV  31  4/5  + 85 

^'  25^2  •  11 

5.  8  days.  6.     16  days. 


82  ANSWERS  TO  QILLET'S  ALGEBHA. 

EXERCISE  CVIII. 

I.  a:  1=  3,          y  =  ±b,  2.    x  =  7/2,        y=±  5/2, 
x=  -3,     y  =  ±5.  x=  -  7/2,  y=±  5/2. 

3.    x  =  ZV%^    y—±2V5,  4.    x='dm.—n,y=±{m-\-3n), 

x  =  ~3  i^2,  y=±2V5.  x=n—dm,y=±{m-\-^7i). 

5.    25,  9/16.  6.    -  243,    '^'26^ 

7.    {a^  +  l)'fa~^-^  -  <?2.  8.    (^x  +  8Z^)(x  -  2«). 

EXERCISE  CIX. 

I.  a:  =  2,             y  =  ^,  2.     X  =  4:,        y  =  —  e, 
x=  -  7/5,  2/  =  49/5.  X  =  -  6,  y  =  4:. 

Z,    X  =z  5,        .^  =  3,  4,    a;  =  5,        y  =  9^ 

x=  —  3,  y  =  —  5.  X  =  —  1,  y  =  3. 

5^    X  =  G,        y  =  ^f  Q,    X  =  6,  y  =  3, 

x—  —  S,y—  —  i).  X  =  3,  y  =  6. 

7.    X  =  a  -{-  1,  y  —  a,  8.    x  =  4:,  y  =  6. 

x=  —  a,     y  =  —  a  —  1. 

9.    X  =  —  1,  y  —  —  1,  10.    X  =  4,            y  =  12, 

X  =  1/2,  y  =  2.  x  =  -  36/7,  ?/=-12/7. 

II.  (±  2)",  (-14/3)"/2.  12.     (3a-2)(5a;  +  2^>). 

EXERCISE  ex. 

I,    X  =  7,  y  =  Q,  2.    X  =  8,  y  =  3, 

X  =  6,  y  =  7.  x  =  3,  y  =  8. 

Z,    X  =  5,  y  =  2,  4.    rr  =r  3,  2/  =  7, 

x=  2,  y  =  5.  X  =  7,  y  =  3. 

6.    X  =  1,       y  =  ^>  Q,    X  =  '2,  y  =  S, 

X  =  —  b,  y  =  —  1.  X  =  —  8,    y  =  --  2. 

1^    X  —  2,  y  —  —9,  %,    x=  —  Q,    y  =zl2f 

X  =  9,  y  —  —  2.  x=  —  12,  ?/  =  6. 

9.    a;  =  7,        2/  —  4,  10.    a:  =  5,  ?/  =  3, 

a;  —  —  4,  ?/  =  —  7.  ^  =:  3,  2/  =  5. 

II.  X  =  Q,       «/  =  4,  12.    X  =  5,  y  =  8, 
x=  — 4,  «/  =  —  6.  ic  =  8,  ^  =  5. 


AN8WEBS  TO  GILLET'S  ALGEBRA.  33 

13.  X  =  Qi,  y  =  3,  14.     ^  r=  9,  2/  =  "^j 
:c=  3,  ?/=.6.  :k  =  _  7,  2/-  -9. 

15.    x  =  h -\-a,  y=a—h,   16.  x=±(2a—b),  y=±{a—2b), 
x=b—ay      y=—a—b.     x=±{a—2b),y=±(2a—b). 

EXERCISE    CXI. 

1.    x=  ±  4:,    y  =  ±1,  2.    X  =  ±  8,  y  =^  ^^  5, 

x=^  ±  14:,  y=  ^4:.  x=  ±3,  y=  ±  5, 

3.  x=  ±Q,  y=  ±2.  4.     x=  :i^9,  y=^  ±4. 

6.  X  =  ±  4,         y  —  ±  b,       Q.     X  =  ±2,      y  —  ±4, 

x  =  ±^V^,y=±V^.  x=  ±V2,  y  =  ±3V2. 

EXERCISE  CXIlo 

1.    3  and  5.  2,    4  and  7.  3.    5  and  9, 

4.  4  and  10.  5.    3  and  4.  6.    3  and  7. 

7.  2  and  3.  8.    1  and  2. 

9.    Cows  30  dollars  apiece  and  sheep  3  dollars  apiece. 

10.    13.  11.    ^^:rj2' 

12.    {4x  -  dy'^)(dx^  -  2y).      is.     - 

14.  25.         15.    4  4/2".        16.    ^ 


26b^  -  84ac 

3Qd' 

' 

b 

17. 

4+|/2. 

III. 

3. 

4, 

8;  13,  1. 

6. 

9, 

8,3. 

EXERCISE    CXIII 

1.  2,  3.  2.    1,  10;  14,  2. 

4.  1,  11.  5.    7. 

7.  5,  6,  7.  8.    4,  2,  7.- 

9.  3,  11,  1;  7,  4,  1;  2,  8,  2;  6,  1,  2;  1,  5,  3. 

10.  1,  5,  2;  3,  1,  4;  2,  3,  3. 

11.  a;  =  4  +  13i?,  2/  =  1  +  '^i>- 

12.  a;  =  lli>  —  2,  y  =  ^p  —  2. 

13.  8,  7.  14.    64,  44. 

15.  liy  using  the  7-inch  five  times  and  the  13-inch  once. 


34 


ANSWERS  TO   GILLETS  ALGEBRA. 


16.  By  using  6  four-pound  weights  and  3  seven-pound 
weights. 

17.  By  using  the  fifty-  and  twenty-cent  pieces  respec- 
tively 1,  17;  3,  12;  5,  7;  or  7,  2. 

18.  By  using    the   half-dollars,  quarter  -  dollars,  and 
dimes  respectively  1,  18,  1;  4,  10,  6;  or  7,  2,  11. 

19.  5  pigs,  10  sheep,  and  15  calves.         20.    92,  90. 
21.    19/9,  2/5;  10/9,  7/5;  or  1/9,  12/5. 


EXERCISE   CXIV. 

Z.    X>  2i. 

4.  x>m. 

5.    ^>4|. 

6.    X  >  3.9,  y  >  4.9. 

EXERCISE  CXVII. 

1. 

151  :  208. 

2.    6  :  11.               3.    589  :  1008. 

4. 

x^  -y^'.x 

—  y.                   6.    x^  —  y^  \x  — 

■y- 

6. 

x""  -%f\x 

-  y.          7.     144  :  125.           8 

.    15  :  8. 

9. 

0,  4,  16,  oc 

) ,  -  32.    10.     -  If                11 

.    18. 

12. 

385,  660. 

13.     11.                         14 

.    5  :  37. 

15. 

5  :  6  or  -  : 

3  :  5.         16.    9  :  7,  or  -  8  :  3. 
EXERCISE  CXVill. 

17.    5. 

1. 

ah 

c  ' 

bb                            1 
2.     y.                    3.     -. 

9 
4.    -. 

c 

5. 

3i. 

4.a 

n 

6.             O*.                 7.           /7      ■    \« 

^                   m{b—a) 

*•    'd{c-h)' 

9. 

2. 

10.     -4.            11.     -2i.      12.    6. 

13. 

2i. 

14.    1/2.             15.     -3/14. 
EXERCISE  CXIX. 

1. 

13i 

2.    2|.               3.    3.6. 

4.     16. 

5. 

1256.64. 

6.    523.5  cu.  ft.           7. 

4752  cu.  ft. 

8. 

2^  cu.  ft. 

9.    18  miles.               lo. 
EXERCISE  CXX. 

15/32. 

1. 

2.9805. 

2     1.7686.          3.    0.3766. 

4.    2.5119. 

5. 

1.6990. 

6,    3.4771.          7.    4.6021. 

8.    0.3010. 

AN8WEBS  TO   GILLET'S  ALGEBRA. 


35 


9. 
13. 

6.8451.       10. 
0.3923.       14. 

4.4571.        11.    1.2121 
0.9034. 

EXERCISE  CXXI. 

). 

12.    3.5538 

1.    862. 
4.    7665. 
7.     .2864. 

2.     .366. 
6.     3.9645. 
8.     .09034. 

EXERCISE  CXXII. 

3. 
6. 
9. 

.0988. 

.006823. 

6.42285. 

1.    6.42221. 
4.     10.3701. 
7.    8.3010. 

2.     6.4024. 
5.     11.1025. 
8.     13.0969. 
EXERCISE  CXXIII. 

3.     6.5383. 
6.     12.0969. 
9.    14.0458. 

1. 

4. 

7. 
10 
13. 
16. 
19. 

172. 
.000406. 

-  2340.52. 
.000000636. 

-  378.45. 
2.3388. 
23.2578.     20. 

2.      .677.                      3. 

5.     .0114289.            6. 

8.      118.916.              9. 
11.     4.326.                 12 
14.      7.12.                       15 
17.      —  .006535.        18 
.8834.     21.      .15811. 
EXERCISE  CXXIV. 

22 

-  127.205. 

-  1299.39. 
645300. 
1.71. 
.07852. 
.2475. 

5.      -  .70214 

1. 

3.9073. 

2.    3.4022. 

3. 

1.4999. 

4. 
7. 

2.7871. 
6. 

5.    2.1683. 
8.     5/2. 

6. 
9. 

18346. 
-1/3. 

1. 
6. 
9. 

1.                  2. 
m/p.           6. 
2«.            10. 

EXERCISE   CXXV. 

00  .                3.     a/h. 
0.                  7.     -  10/7. 
5z*.            11.     -  3/2. 
EXERCISE   CXXVI. 

4.    h/a. 
8.    -9/4. 
12.     -2. 

1. 

3. 

7. 
11. 
13. 

64;  78;  -  75 
11.                 4. 
3i.                8. 
9th.            12. 

8/9,  7/9,  6/9. 

;8.               2.    52;  83;  - 
7.              5.    2h 
0.               9.     19  th. 
10,  12,  14,  .  .  .  52. 

,  .  .  .  1/9. 

-14;  55;  -19f. 
6.     1/6. 
10.     16th. 

36  ANSWERS  TO   QILLET'S  ALGEBRA. 

14.  4a  —  bh,  3a  —  4:b,  2a  —  'db  ,  .  .  —  5a  +  ib. 

15.  d  =  4:,  a  =  2.      16.    d=  —  3,  a  =  21.      17.     -  28f. 

EXERCISE  CXXVII. 

1.  624.        2.  187.         3.  255. 

4.  810.         5.  0.  6.  357. 

7.  1/2  {n^  +  37^2).       8.  n{a  +  ^')2  -  oi(n  -  l)ab, 

9.  80.         10.  1941.        11.  1080. 

12.  1160.     13.  8  +  12  +  16  +  .  .  .  +  76.  /S'  =  680. 

14.  12j\  +  14if  +  16f f  +  .  .  .  97|f .  ^  =  2200. 

15.  8729.  16.  41832. 

EXERCISE    CXXVIII. 

1.  603.       2.     3375.       3.     13.        4.     33.        5.     10  days. 

6  8  clays.  7.     ±5.  8.     ±  2^.  9.     9  days. 

10.  50500  yards.       11.     $5195.       12.     ±  20,  ±  30,   ±  40. 

13.  ±  8,  ±  12,  ±  16,  ±  20.     14.     =F  4,  ±  2,  ±  8,  ±  14. 

EXERCISE    CXXIX. 

1.      10,  50.  2.      ±  12,  -48,  ±  192. 

3.  -  15,  45,  -  135,  405. 

4.  ±  .6,  .12,  ±  .024,  .0048,  ±  00096 

6.  1/3,  2/3,  4/3,  8/3,  16/3,  32/3. 

EXERCISE  CXXX. 

1.  19680.      2.  -9840.      3.  1281/512. 

4.  191i.       5.  -682.       6.  53144/2187. 

7.  -463/192.   8.  64/65.  9.  27/58. 
10.  .999.       11.  1/2.        12.  4. 

13.  6,  24,  96,  384,  1536.   14.  -  12,  36,  -  108. 

15.  24,  60,  150;  or  27,  63,  147. 

EXERCISE  CXXXI. 

1.  5/33.   2.  5/27.   3.  44/111.   4.  3/7.   5.  1/77. 
6.  4/5.    7.  52/165.     8.  7/60.     9.  143/740. 


ANSWERS  TO   OILLETS  ALGEBRA  37 


EXERCISE   CXXXII. 

1. 

$4159.09.                   2.     i?1153.94. 

3. 

$897.00. 

4. 

5?  yrs.                        5.     $403.90. 

6. 

.04. 

7. 

14  yrs.  2  mo.  12  da.    8.     17^  yrs. 

9. 

$6785.71 

10. 

$6000.                       11.     $3246.42. 

12. 

$437.50. 

13, 

451.33. 

EXERCISE  CXXXIII. 

1.    4  and  6.                       2.     1/2  and  2/7. 

3.    1/10.                           4.     6i  8i, 

m. 

EXERCISE  CXXXIV. 

1. 

1.2.1.                                   2.     1.3.3.1 

.. 

3. 

1.4.6.4.1.                           4.     1.5.10. 

10.5.: 

1. 

5. 

1.6.15.20.15.6.1.            6.     1.7.21. 

35.35 

.21.7.1. 

7. 

1.8.28.56.70.56.28.8.1. 

8.  1.9.36.84.126.126.84.36.9.1. 

9.  1.10.45.120.210.252.210.120.45.10.1. 

EXERCISE   CXXXV. 

1.  a^^  X  {:ix'y  =  16«iV2,      2.    32  X  (-  ay^  ~  -  S)a^\ 

3.  (5«3)4(_  7^^3)3  ^  _  214375^%^ 

4.  5Vx-J--«V.       5.     (2^)^(-^)-l^ 
6.    4^X^=1. 

EXERCISE  CXXXVI. 

2.  a^  -  8t«^a;  +  28««a;2-56«5:x;3  _^  TO^?*^*-  56A-5  +  28a2a;6 

—  Sax^  -f  a;^. 

3.  1  +  9:?;  +  36:?;2  _^  34^3  _|_  126a:*  +  126^;^  +  Mx^  +  Ux'^ 

+  9^'S  4-  x\ 

4.  :r^  -  15a;4  +  90.r3  -  270a;2  +  405a;  -  243. 

5.  81a;4  +  'ZUx^y  +■  216a;y  +  %Qxif  +  16?/^ 

6.  32a;5  -  SOx^ij  +  80^:3^2  _  49^2^3  _|_  i()^^4  _  ^5^ 

7.  1  -18a2  +  I35a^  -  540rt«  +  1215a^  -  1458^1^  +  729a^l 


38  ANSWERS  TO   GILLET'S  ALGEBRA. 

8.  l-7a;?/+21a;y-35^-y+35:z;y-21a.-y+  "ixSj^-x^if. 

9.  729««  -  972a^  +  540«^  -  IQW  +  ?|^'  _  ^^  +  ^^. 

^°-    729  +    27  "^     3     "^  "    "^  4a:2  +  8^^*  +  g4^,6- 
12.    w?~^  —  6m~^/W  +  15?/?r2'M'*  —  207n~^/hi^  +  15wr^;i^ 

14.  a^2  _^  20a9a;V2  +  150A  +  500A3/2  ^  Q2bx^ 

15.  «^  +  16«29/6  -^  96ftii/3  -I-  25Gft5/2  4_  256^^3. 

16.  x^  +  15a;i2/5«/-2/5-j-90a:V5;i/-4/5-f  270x6/5^-«/5+405:?;3/5^-8/3 

+  243?/-l 

17.  ttV2^-lV3_|-7a5/2^-10/3_|_21^3/2^-2_^  35«V2^-2/3_|_35^-l/2^2/3 

+  21a-3/2^2  _|_  7^-5/2^10/3  _^  a-y^^y. 
10   45   120   210   252   210   120   45 

10   1 

3o' 


ic^   a;^ 


EXERCISE    CXXXVII. 


1.     -  35750a;^«.  2.     -  112G402;9.  3      -  312a;2. 

.n  '^'^  ll'^O   ,,,  10500 

81  x^ 


70:ry»  ^  ^^     2a;4  +  24a;2  +  8.     9.     140  V^. 

10.    2(3G5  -  'SQ3x  +  63.^2  -  x^).  li.     252. 

189^^^  _  21a^  J_ 

12.         g     ,         ^g  .  13.     ^g. 

EXERCISE   CXXXVIII. 

1.  aV4  _  l/4«-3/4^  _  3/32«-V4a;2  _  r/128«-"/V 

-  77/2048rt-iVV. 

2.  a^/2  +  3/2aV2a;  +  3/8«-V2a;2  _  1/16^,-3/2^^3 

-f  3/128a-V2a;4. 


ANSWERS  TO   GILLET'S  ALOEBIiA.  39 

3.  1  +  4.«  +  10.f2  -f  mx^  4-  'dbx\ 

4.  1  -  Ix  +  28.^2  _  84a;3  +  210a;^ 

6.    3V4  _  _J_  X l-x^ \—x^ '^-l :^, 

2V27"         8^3'^  l^h'^  128^3^ 

6.     1  +  1/3:^;  +  2/9:^2  4-  14/81a;3  +  35/243^;^ 

1        1      .     3     o         11     3    ,      44     , 
'■     ^-r  +  ^^^-T2^^    +-625^- 

8.  x-^  -  ^x-'^ij  +  IQx-hf  -  Ux-y  +  256a;-iy. 

9.  «-i  +  1/2^-5:^-1/2  _j_  b/8a-^x'^  +  15/16«-%-3/2 

+  195/128«-%-2 

EXERCISE  CXXXIX. 


1. 

4. 

7. 

10. 

8648640. 
720. 
27720. 
240. 

2.    259459200.         3. 

5.     181440.                6. 

8.    840.                      9. 

11.     96.                       12. 

EXERCISE  CXL. 

5040. 
90720. 

480. 
9^89180. 

1. 
4. 

70. 
1512000. 

2.     10080. 

5.     178378200. 

EXERCISE  CXLI. 

3.     5250. 
6.     455. 

1. 
2. 
3. 
4. 
5. 
6. 

7. 

a;3  _   Qx^  _|_ 

x^  -  4.^•3  - 

6a;^  -  1L^•3 
9.T^  4-  30a;3  - 
20:^^4-21a;3 

3,  -r. 

11a;  -  6  ==  0. 

19a;2  +  46:^  +  120  =  0. 

x^  -  6x  =  0. 

-  48x2  —  19x  +  12  =  0. 

-47:^2-  120.r+  144  =  0. 

-  240^:2  -  Vdx  +  12  =  0. 

8.     -  2,   |. 

9. 

-  2±  i^. 

10.    ^(-3± 

V-7). 

11. 
13. 

^(-1±V 
3,  -4. 

-  8).             12.  2,  2. 
14.     -  2,  4.               15. 

-7,8, 

40  ANSWERS  TO   GILLETS  ALGEBRA. 


EXERCISE  CXLII. 

and       ,   „.  2.    ^      and— 


3.   • z     and r— r.         4.    ^     and 


re  —  4                :»+3             '    X  —  8  X  -}-  6' 

6.    ^ — r-p>  and -.  6.    ^ 1  and   — 


2a;  +  3  a;  —  5  *    3a;  —  4  2a;  —  6' 

EXERCISE    CXLIII. 

3  4 


1. 


a;  —  3   'a;  —  4      a;  —  5 


3  2       ,       5 


3. 


2a; +  2       a;-3'a;  +  3 
1.4  1 


2a; -1    '3  +  a;      3-a; 

1                4,7 
4.   irrz T^-z -0  + 


2(a;  -  1)      a;  -  2  ^  2(a;  -  3)  • 

3  5  1 

5. 


4(a;  +  3)       8{x  +  5)       8(x  +  1)' 
1  7  13 


6-    1o/^_J_1^-^ ^^4- 


12(a;  +  1)       3(a;  -  2)  ^  4(a;  -  3)* 


EXERCISE   CXLIV. 

a;-  2 


3. 


*• '  3(a;  +  1)       3(a;^  -  a;  +  1) 

7  5a;  -  3 

^'    x  —  l+a;2_|_  x-\-l' 

5a; +6      _       3a;  -  4 
a;'^  +  a;+l       a;^  —  a;-|-l* 
1  4a; -8 

*•     5(a;+2)  +5(a;2+  1)* 

11 
^-     2(a;2  +  1)  "^  H^  ~  1)' ' 


ANSWERS  TO  OILLETS  ALGEBRA.  41 

EXERCISE   CXLV. 

1.  1  -  bx  +  nx"  -  86.^3  +  M^xK 

2.  2  -  Ix  +  28a;2  -  ^Ix^  +  322^;^ 

3.  3  -  19:^2  +  95a;4  -  475^6  +  2375^8. 

4.  2  -  11:?;2  _^  44^4  _  1^(5^6  _|_  704^:8. 

3       ,     9    ,    ,     27    ,    ,     81     .    ,     243     9 

2       ,  4    „    ,     8    ,    ,    16    „    ,     32    3 
«•    3-"+r"   +^^+ 81^+243^  • 

EXERCISE   CXLVI. 

.    ,    1       >  3    2        3     ,   ,      3      , 

1.  l  +  2^+8^'^-r6^+l28^- 

2.  1  -  a;  4-  -a;2  -  -2;3  +  -x^. 

3.  1+3^-^-^  +  81^'^ -^3^^. 


EXERCISE  CXLVII. 

1  12.^3  13       4 

1.  ^=2^-8^^+16^  -128^- 

2.  X  =  y  —  y'^  -^  1/  —  y^. 

3.  a;  =  2/+2/'  +  2/+8/  +  ..- 

EXERCISE  CXLVIII. 

1111  111111 

1-     1  +  8+1+5*  ^-    1+3+1+3+1+3 

3    3  +  1      1      i      i      i      ^      i 

^-       ^1+1+1+1+3+2+2 

1^111111 

^•.1  +2+1  +  2  +1  +2  +1+2* 

5    2+1      1      1      1      i      i      1 

^-       ^3  +2  +1  +3  +2  +1  +2 

i_j_l       11^       11111 

6-  i  +  i+i+i+r+i+i+i+1+3-    . 


42  ANSWERS  TO   GILLETS  ALGEBliA. 

1  4.L       1       L       L       1       1       1 

^-      +3"  +  l  +  a +r  +  3 +1+3* 
L    1    L    1    L    i 

*■  2+3+4+5+6+7* 

9.  2/1,  13/6,  15/7,  28/13,  323/150,  674/313. 

10.  1/2,  2/5,  7/17,  9/22,  25/61,  159/388. 

11.  3/1,  10/3,  13/4,  36/11,  85/26,  121/37,  1174/359. 

12.  1/2,  3/7,  4/9,  19/43. 

13.  1/4,  7/29,  8/33,  39/161,  47/194. 

1   1   L   1   1   ^.lL     .76     1 
I*'-  2  +  3  +  3  +  3  +  1  +  1  "^  7  +  .  .  . '  175 '  262325' 

_\ 

231700* 

18.  -  +  2^  _^  2  +  1  +  1  +  4  +  1  +  1  +  19  +  .  .  . '  71  ' 

1__    _1_ 

103589'  98548* 

EXERCISE  CXLIX. 

1   2^1   i   1   1   1   1   1   1 

'•   -^1+1  +  1+4+1  +  1  +  1  +  4+.  ,, 

^'       "^1+1  +  1+1  +  6  +  1+1+1+1+6+.., 


i  1         „  .  1   1   1   1 

2+2 
1   1 


^'     "^"^2+2...    *•  ^  +  2  +  4+2+4+.. 


^-  ^+8+8+... 

,1111_11 
®-   +  2  +  i_^3_j_i4_2  +  8 

1   1_   1   1   1   1 

+  2  +  1  +  3  +  1  +  2  +  8  +  .• 

7.   1^2-1.     8.   1^6-1.     9.  1/5(2  1^39-9). 


COMPLETE   LIST 

OF 

HENRY   HOLT  &   CO.'S 

EDUCATIONAL  PUBLICATIONS. 

All  prices  art  net  except  those  marked  withan  asterisk  (*),  which  are  retail. 
All  books  bound  in  cloth,  unless  otherwise  indicated. 


SCIENCE.  CATALOGUE 

PRICE  PAGE 

Allen's  Laboratory  Physics,  PupiVs  Edition %    80  2 

T^&  s,2i.xn^.  Teacher'' s  Edition   100  2 

Arthur,  Barnes,  and  Coulter''s  Plant  Dissection 120  3 

Barker's  Physics.  A  dvanced  Course 3  50  4. 

Beal's  Grasses  of  North  America.     2  vols 175 

'^&%%Qy''S^Q\.2L.\-\Y,  Advanced  Course   220  6 

'Y\\&%3i'SX\&,  Briefer  Course 108  6 

Black  and  Carter's  Natural  History  Lessons 50  8 

Bumpus's  Laboratory  Manual  of  Invertebrate  Zoology 100  8 

Cairns's  Quantitative  Analysis  i  60  176 

Hackel's  True  Grasses  (Scribner)    .  *i  50  8 

Hall  and  Bergen's  Physics  (A'^>',  50  cts.) 125  9 

Hall's  First  Lessons  in  Physics   .    65  177 

Hertwig's  General  Principles  of  Zoology 178 

Howell's  Dissection  of  the  Dog 100  10 

Jackman's  Nature  Study   120  n 

Kerner's  Natural  History  of  Plants.  With  16  colored  plates,  looocuts.  4  Pts.  15  00  179 

Macalister's  Zoology 80  12 

MacDougal's  Experimental  Plant  Physiology i  00  179 

Macloskie's  Elementary  Botany i  30  12 

McMurrich's  Invertebrate  Morphology 4  cx)  iBd 

McNab's  Botany 80  12 

MsLTtin's  The  Huma.n  Body,  A d7>anced  Course 220  13 

The  same.  Briefer  Course i  20  13 

The  same,  Elementary  Course 75  15 

The  Human  Body  and  the  Effects  of  Narcotics i  20  14 

Newcomb  and  Holden's  Astronomy,  ^(f7'««<r«^^  C<7Mrj^ 200  16 

"Yh^  %Ayci^.  Brief er  Course i  12  16 

Noyes's  (W.  A.)  Elements  of  Qualitative  Analysis 80  17 

Packard's  Zoology,  ^^z/rtAzc^^/ Ct'z^r.f^ 240  18 

T\\t.?,3i'caQ.,  Briefer  Coiirse i    12  i3 

The  same,  Ele^nentary  Course 80  19 

Entomology  for  Beginners *i  40  20 

Guide  to  the  Study  of  Insects  *5  00  20 

Embryology *2  50  20 

Remsen's  Chemistry,  .<4£/z/rt«c^^  C^«rj^ 280  21 

The  same.  Briefer  Course 1  12  21 

The  same.  Elementary  Course 80  23 

Laboratory  Manual  (for  Elementary  Course) 40  23 

Remsen  and  Randall's  Chemical  Experiments  (for  Briefer  Course) 50  181 

Scudder's  Butterflies ""i  50  24 

Brief  Guide  to  Commoner  Butterflies *i  25  24 

Life  of  a  Butterfly *i  00  24 

Sedgwick  and  Wilson's  General  Biology,  New  Edition 175  i8i 

Underwood's  Native  Ferns 100  26 

Williams's  (,G.  H.)  Elements  of  Crystallography i  25  26 


Complete  List  of  Henry  Holt  &-  Co.'s 


CATALOGUE 

PRICE  PAGE 

Williams's  (H.  S.)  Geological  Biolog-y  $280  182 

WoodhuU's  First  Course  in  Science  :   Book  0/ Experiments 50  183 

Text-book 65  183 

Zimmermann's  Botanical  Microtechnique  — z  50  184 

MATHEMATICS. 

Gillet's  Elementary  Algebra 186 

Euclidean  Geometry 186 

Keigwin's  Class-book  of  Geometry 187 

Newcomb's  School  Algebra  (^^j>',  95  cts.) 95  29 

Algebra  for  Colleges  (A'^jj',  $1.30) i  30  29 

Elements  of  Geometry   120  29 

Plane  and  Spherical  Trigonometry 160  30 

Trigonometry,  separate i  20  30 

Mathematical  Tables i  10  30 

Essentials  of  Trigonometry 1  00  30 

Plane  Geometry  and  Trigonometry i  10  30 

Analytic  Geometry 120  31 

Differential  and  Integral  Calculus 150  31 

Phillips  and  Beebe's  Graphic  Algebra i  60  32 

HISTORY  AND    POLITICAL    SCIENCE. 

Doyle's  History  of  the  United  States 100  36 

Duruy's  Middle  Ages i  60  33 

Modern  Times  to  1798 160  189 

Fleury's  Ancient  History  told  to  Children..   70  34 

Freeman's  General  Sketch  of  History i  10  35 

Fyffe's  History  of  Modern  Europe  :  Volume  I.  1792-1814 *2  50  37 

Volume  n.  1814-1848 *2  50  37 

Volume  HI.  1848-1878 *2  50  37 

Gallaudet's  Manual  of  International  Law 130  37 

Gardiner's  English  History  for  Schools 80  38 

Introduction  to  English  History 80  39 

Gardiner  and  MuUinger's  English  History  for  Students i  80  39 

Hunt's  History  of  Italy ..  80  36 

Johnston's  American  Politics 80  43 

H  istory  of  the  United  States 1  00  40 

Shorter  History  of  the  United  States 95  42 

Lacombe's  Growth  of  a  People 80  44 

Mac  Arthur's  History  of  Scotland 80  36 

Porter's  Constitutional  History  of  the  United  States i  20  44 

Roscher's  Principles  of  Political  Economy.    2  vols *7  00  44 

Sime's  History  of  Germany 80  36 

Sumner's  Problems  in  Political  Economy 1  00  44 

Thompson's  History  of  England 88  35 

"Walker's  Political  Economy,  Advanced  Course 2  00  45 

The  same,  Briefer  Course i  20  46 

The  same  Eletnentary  Course i  00  46 

Yonge's  History  of  France 80  36 

Landmarks  of  History  :  Ancient  History 75  48 

Mediaeval  History 80  48 

Modern  History 103  48 

PHILOSOPHY. 

Baldwin's  Psychology.    Vol.  I.  Senses  and  Intellect 180  49 

Vol,  II.  Feeling  and  Will 200  50 

Elements  of  Psychology .   150  51 

Descartes,  Philosophy  of  (Torrey) 150  56 

Falckenberg's  History  of  Modern  Philosophy 3  50  52 

Hume,  Philosophy  of  (Aikins) i  00  192 

Hyde  s  Practical  Ethics 80  53 


Educational  Publications  iii 


CATALOGUE 

PRICK  PAGB 

James's  Principles  of  Psychologry.    2  vols $480  54 

VsychoXogy,  Br ie/er  Course  160  55 

Kant,  Philosophy  of  (Watson) i  75  56 

Locke,  Philosophy  of  (Russell) 100  56 

Paulsen's  Introduction  to  Philosophy  (Thilly)  350  191 

Reid,  Philosophy  of  (Sneath) i  50  56 

Spinoza,  Philosophy  of  (Fullerton) 150  192 

Zeller's  History  of  Greek  Philosophy , 140  57 

MISCELLANEOUS.   (In  English.) 

Banister's  Music 80  58 

Champlin's  Cyclopaedia  of  Common  Things.     Cloth *2  50  59 

The  same.     Half  Leather — *3  00  59 

Cyclopaedia  of  Persons  and  Places.     Cloth *2  50  60 

The  same.     Half  Leather  *3  00  60 

Catechism  of  Common  Things 48  6i 

Young  Folks'  Astronomy 48  61 

Champlin  and  Bostwick's  Cyclopaedia  of  Games  and  Sports *2  50  6i 

Cox's  Catechism  of  Mythology .- 75  62 

Davis,  King,  and  Collie's  Governmental  Maps  — 30  62 

"White's  Classic  Literature 160  62 

>Vitt's  Classic  Mythology, 100  62 

ENGLISH   LANGUAGE   AND   LITERATURE. 

Bain's  Brief  English  Grammar  (/To',  40  cts.) 40  63 

Higher  English  Grammar 80  63 

English  Grammar  bearing  upon  Composition i  10  63 

Baker's  Specimens  of  Argumentation.     Modern.     Boards 50  193 

Baldwin's  Specimens  of  Prose  Description.     Boards 50  194 

Boswell's  Life  of  Dr.  Samuel  Johnson  (abridged) *i  50 

Brewster's  Specimens  of  Prose  Narration.     Boards 50  195 

Bright's  Anglo-Saxon  Reader i  75  64 

ten  Brink's  History  of  English  Literature  :  Volume  L  To  Wyclif *2  00  65 

Volume  IL  (Part  L) *2  00  65 

Clark's  Practical  Rhetoric 100  66 

Exercises  for  Drill.     Paper  35  66 

Briefer  Practical  Rhetoric 90  67 

Art  of  Reading  Aloud 60  67 

Coleridge's  Prose  Extracts.     (Beers.)     Boards 30  196 

Cooks  Extracts  from  Anglo-Saxon  Laws.     Paper 40  68 

Corson's  Anglo-Saxon  and  Early  English 160  68 

De  Quincey's  English  Mail  Coach  and  Joan  of  Arc.     (Hart.) 30  197 

Ford's  The  Broken  Heart.     (Scollard.)     Cloth 70  197 

The  same.     Boards 40  197 

Hardy's  Elementary  Composition  Exercises  80  68 

Johnson's  Chief  Lives  of  the  Poets.     (Arnold.) 125  68 

Rasselas.    (Emerson.)    Cloth 70  198 

The  same.     Boards 40  198 

Lamont's  Specimens  of  Exposition.     Boards 50  199 

Lounsbury's  History  of  the  English  Language i  12  203 

Lyly's  Endymion.    (Baker.)    Cloth 125  199 

The  same.     Boards  85  199 

Macaulay  and  Carlyle:  Croker's  Boswell's  Johnson  (Strunk.)     Boards.        40  2cxd 

Marlowe's  Edward  H.  (McLaughlin.)    Cloth —    70  201 

The  same.     Boards  40  201 

McLaughlin's  Literary  Criticism 100  70 

Nesbitt's  Grammar-Land *i  00  70 

Newman:     Selections.     (Gates.)    Cloth 90  201 

The  same.     Boards 50  201 

Pancoast's  Representative  English  Literature 160  204 

Introduction  to  English  Literature 125  206 

Sewell's  Dictation  Exercises 45  77 

Shaw's  English  Composition  by  Practice 75  76 

Siglar's  Practical  English  Grammar 60  77 


iv  Complete  List  of  Henry  Holt  &-  Co.'s 


Smith's  Synonyms  Discriminated 

Taine's  History  of  English  Literature. 


CATALOGUE 

PRICE  PAGE 

*|2  25  77 

*i  25  77 

The  SAWie,  Abridged.     Class-room  Edition.    (Fiske.) i  40  77 

GERMAN  LANGUAGE. 

BlackweU's  German  Prefixes  and  Suffixes 60  78 

Bronson's  Colloquial  German  {Key,  65  cts.) . .  65  79 

Easy  German  Prose.    Se&aXso  Andersen,  Grimm,  and  Haujff^  125  213 

Fischer''s  Practical  Lessons  in  German 75  79 

Elementary  Progressive  German  Reader  , 70  215 

Wildermuth's  Der  Einsiedler  im  Walde 65  80 

Hillern's  Hoher  als  die  Kirche 60  80 

Harris's  German  Reader 100  218 

Heness's  Der  neue  Leitfaden i  20  81 

Der  Sprechlehrer  unter  seinen  Schiilern i  10  81 

Huss's  Conversation  in  German i  10  81 

Jagemann's  German  Prose  Composition 90  82 

Elements  of  German  Syntax 80  83 

Joynes-Otto:    First  Book  in  German.     Boards  30  84 

Introductory  German  Lessons 75  84 

Introductory  German  Reader  95  84 

Translating  English  into  German  {Key,  80  cts.) 80  84 

Kaiser''s  Erstes  Lehrbuch 65  85 

Keetels' Oral  Method  with  German 130  85 

Klemm's  Lese- und  Sprachbiicher.  Kreis      I.    Boards 25  86 

"        II.    Boards 30  86 

"         "     (WilhVocab.) 35  86 

"      III.    Boards 35  86 

"     (WithVocab.) 40  86 

"      IV.    Boards 40  86 

"        V.    Boards 45  86 

"      VI.    Boards 50  86 

"    VII.    Boards     60  86 

Geschichte  der  deutschen  Literatur  (Kreis  VIII.) 1  20  86 

Otis's  Elementary  German 80  87 

Introduction  to  Middle  High  German   100  88 

Otto's  German  Conversation  Grammar  {Key,  60  cts.) i  30  89 

Elementary  Grammar  of  the  German  Language 80  90 

Progressive  German  Reader.     Half  roan  i  10  90 

Pylodet's  New  Guide  to  German  Conversation 50  91 

Schrakamp  and  van  Daell's  Das  deutsche  Buch  65  91 

Schrakamp's  Erzahlungen  aus  der  deutschen  Geschichte 90  122 

Spanhoofd's  Das  Wesentliche  der  deutschen  Grammatik 60  92 

Sprechen  Sic  Deutsch  ?     Boards 40  92 

Stern's  Studien  und  Plaudereien,    First  Series.    New  Edition  1  10  227 

"           "                "           im  Vaterland.     Second  Series i  20  94 

Teusler's  Game  for  German  Conversation.     Ninety-eight  Cards  in  a  Box  80  95 

Thomas's  Practical  German  Grammar i  12  228 

Wenckebach  and  Schrakamp's  Deutsche  Grammatik  i  00  96 

Wenckebach's  Deutsches  Lesebuch   80  97 

Deutscher  Anschauungs-Unterricht i  10  97 

Die  schonsten  deutschen  Lieder 120  106 

Whitney's  Compendious  German  Grammar  {Key,  80  cts.) 1  30  98 

Brief  German  Grammar 60  99 

German  Reader  in  Prose  and  Verse 150  100 

Introductory  German  Reader 100  229 

German  and  English  Dictionary 2  00  loi 

Whitney-KIemm:  German  by  Practice 90  102 

Elementary  German  Reader 80  102 

W^illiams's  Introduction  to  German  Conversation 80  102 

V^itcomb  and  Otto's  German  Conversation 50  91 

GERMAN   LITERATURE. 

Andersen's  Bilderbuch.     Vocab.    (Simonson.)     Boards 30  rri 

Die  Eisjungfrau  und  andere  Geschicliten.  (Krauss.)   Boards  30  iii 

Ein  Besuch  be!  Charles  Dickens      Boards..         25  air 

Stories,  with  Grimm's,  from  Bronson's  Easy  Prose.     Vocab.  90  214 


Educational  Publications 


CATALOGUE 

PRICE  PAGE 

Auerbach's  Auf  Wache  with  Roquette's  Gefrorene  Kuss.  (Macdonnell). 

Boards $    35  i^* 

Baumbach's  Frau  Holde.     (Fossler.)    Poem.     Boards 25  211 

Benedix's  Der  Dritte.     Play.     (Whitney.)     Boards 20  212 

Dr.  Wespe.     P/ay.     Boards 25  118 

Eigensinn.     Piay.     Boards 25  119 

Beresford-\A^ebb's  German  Historical  Reader 90  121 

Carove's  Das  Miirchen  ohne  Ende.     Vocab.     Boards 20  m 

Chamisso's  Peter  Schlemihi.     (Vogel.)     Boards 25  214 

Claar's  Simsoii  und  Delila.     Play.     Paper 25  120 

Cohn's  ijber  Bakterien.     (Seidensticker.)     Paper 30  '^3 

Ebers's  Eine  Frage.     Boards  35  "2 

Eckstein's  Preisgekront.     (Wilson.) 214 

EichendorflF's  Aus  dem  Leben  eines  Taugenichts.     Boards 30  112 

Fouque's  Sintram  und  seine  Gefahrten.     Paper 25  112 

Undine.     Vocab.     (Jagemann.) 80  112 

"            Boards 35  112 

Francke's  German  Literature. 215 

Freytag's  Karl  der  Grosse.     (Nichols.)  75  121 

Die  Journalisten.     Play.     (Thomas.)    Boards 30  118 

Friedrich's  Ganschen  von  Buchenau.     Play.     Paper 35  120 

Goethe's  Egmont.     (Sieffen.)     Play.     Boards 4°  107 

Faust.     Parti.     Play.     (Cook.) 48  107 

Hermann  und  Dorothea.     Poem.    (Thomas.).     Boards 30  107 

Iphigenie  auf  Tauris.     Play.    (Carter) 48  108 

Gorner's  Englisch.    Play.     Paper 25  118 

Gostwick  and  Harrison's  German  Literature 2  00  103 

Grimm's  Die  Venus  von  Milo;  Rafael  und  Michel-Angelo.     Boards 40  112 

Grimms' Kinder- und  Hausmarchen.     Vocab.     (Otis.) .  100  113 

Boards.     (Different  selections  and  notes,  «<?  Vocab.). . .  40  113 

Selections,  with  Andersen,  from  Bronson's  Easy  Prose.     Vocab.  90  214 

Gutzkow's  Zopf  und  Schwert.     Play.     Paper 40  118 

HaufiTs  Die  Karawane.    From  Bronson's  Easy  Prose.     Vocab 75  214 

Das  kalte  Herz.     Boards 20  113 

Heine's  Die  Harzreise.     (Burnett.)     Boards 30  113 

Helmholtz's  Goethe's  Arbeiten.     (Seidensticker.)    Paper 30  123 

Heness's  Kinder-Komodien.     Plays  48  119 

Heys  Fabeln  flir  Kinder.     Vocab.     Boards 30  114 

Heyse's  Anfang  und  Ende.     Boards 25  114 

L'Arrabbiaia.     (Frost.)     Vocab 

Die  Einsamen.     Boards     .    20  114 

Madchen  von  Treppi;  Marion.     (Brusie.)    Boards 25  218 

Hillebrand's  German  Thought  (chiefly  in  Literature)  140  104 

Hillern's  Hoher  als  die  Kirche.     Vocab.     (Whittlesey.)     Boards 25  114 

The  same.     (Fischer.) 60  80 

Jungmann's  Er  sucht  einen  Vetter.    Play.     Paper     25  120 

Klemm's  Abriss  der  Geschichte  der  deutschen  Litteratur 1  20  104 

Klenze's  Deutsche  Gedichte     90  219 

Knortz's  Representative  German  Poems 200  105 

Koenigswinter's  Sie  hat  ihr  Herz  entdeckt.    Play.    Paper 35  120 

Korner's  Zriny.     (Ruggles.)     Play.     Boards 50  108 

Lessing's  Emilia  Galotti.    (Super.)    Play.     Boards..   30  220 


Minna  von  Barnhelm.    Play.    (Whitney.) 48  108 

Nathan  der  Weise.     Play.    (Brandt.)    New  Edition 60  220 

Meissner's  Aus  meiner  Welt.     Vocab.    (Wenckebach.) 75  115 

Moser's  Der  Schimmel.     Play.     Paper 25  120 

Der  Bibliothekar.     Play.    (Lange.)    Boards 40  119 

Miigge's  Riukan  Voss.     Paper 15  115 

Signa  die  Seterin.    Paper 20  115 

Miiller's  Elektrischen  Maschinen.    (Seidensticker.)    Paper 30  123 

Miiller's  (Max)  Deutsche  Liebe.     Boards 35  115 

Nathusius's  Tagebuch  eines  armen  Frauleins.     Paper 25  115 

Nichols's  Three  German  Tales  :    L  Goethe's  Die   neue   Melusine.     H. 
Zschokke's   Der   tote  Gast.      HL  H.   v.   Kleist's  Die  Ver- 

lobung  in  St.  Domingo 60  221 

Paul's  Er  muss  tanzen.     Play.     Paper 25  120 

Petersen's  Princessin  Use.    Boards 30  115 


vi  Complete  List  of  Henry  Holt  &-  Co.'s 

CATALOGUE 

PRICE  PAGE 

Putlitz's  Was  sich  der  Wald  erzahlt.    Paper $    25  116 

Vergissmeinnicht.     Paper 20  116 

Badekuren.     Play.    Paper 25  119 

Das  Herz  vergessen.    Play.    Paper 25  119 

Regent's  German  and  French  Poems.     Boards 20 

Riehrs  Burg  Neideck.     (Palmer.) 30  116 

Der  Fluch  der  Schonheit.     (Kendall.)  25  116 

Roquette's   Der  gefrorene  Kuss,  with  Auerbach's  Auf  Wache.     (Mac- 

donnell.)     Boards 35  117 

Rosen's  Bin  Knopf.    Play.    Paper 25  120 

Scheffers  Ekkehard.     (Carruth.) 125 

Trompeter  von  Sakkihgen.     Poem.    (Frost.) 80  221 

Schiller's  Die  Jungfrau  von  Orleans.    Play.    (Nichols.)    Cloth 60  222 

The  same.     Boards 40 

Das  Lied  von  der  Glocke.    Poem.    (Otis.)    Boards 35  109 

Maria  Stuart.     Play    (Joynes.) 60  223 

Der  Neffe  als  Onkel.    Play.     (Clement.)    Boards 40  no 

Wallenstein.    Play.    (Carruth.) 100  224 

Wilhelm  Tell.     Play.    (Sachtleben.) 48  no 

Schoenfeld's  German  Historical  Prose 80  226 

Schrakamp's  Sagen  und  Mythen 75  226 

Beriihmte  Deutsche ..  85  226 

Simonson's  German  Ballad-book i  10  106 

Storm's  Immensee.     Vocab.     (Burnett.)    Boards 25  117 

Three   German  Comedies  :    Elz's  Er  ist  nicht  eifersuchtig.  Benedix's 
Der  Weiberfeind,  and  MUUer's  Im  War- 

tesalon  erster  Klasse.     Boards 30  119 

Tieck's  Die  Elfen  and  Das  Rothkappchen.     Boards 20  117 

Vilmar  and  Richter's  German  Epic  Tales.     Boards. 35  117 

Wichert's  An  der  Majorsecke.     (Harris.) 20  229 

"Wilhelmi's  Einer  muss  heirathen.    Play.     Boards 25  119 

Zschokke's  Neujahrsnacht  and  Der  zerbrochene  Krug.    (Faust.) 25 

FRENCH   LANGUAGE. 

AUiot's  Contes  et  Nouvelles 100  124 

Hubert's  Colloquial  French  Drill.     Parti 48  125 

The  same.     Part  H     65  125 

Bt  llows's  Dictionary  for  the  Pocket.     Roan  tuck 255  126 

The  same.     Morocco  tuck  310  126 

French  and  English  Dictionary.     Larger-type  Edition 100  126 

Bevier  and  Logic's  French  Grammar 231 

Borel's  Grammaire  Franjaise.     Half  roan 130  127 

Bronson's  Exercises  in  Everyday  French.     (A>^,  60  cts.) 60  23a 

Delille's  Condensed  French  Instruction 40  127 

Eugene's  Students' Grammar  of  the  French  Language i  30  128 

Elementary  French  Lessons 60  128 

Fisher's  Easy  French  Reading 75  128 

Fleury's  L'Histoire  de  France   i  10  128 

Ancient   History 70  128 

Case's  Dictionary  of  the  French  and  English  Languages.     8vo 2  25  129 

Pocket  French  and  English  Dictionary     i8mo 100  129 

Translator 100  129 

Gibert's  French  Pronouncing  Grammar 70  129 

Le  Jeu  des  Auteurs.     Ninety-six  Cards  in  a  Box 80  129 

Joynes"s  Minimum  French  Grammar  and  Reader 75  235 

Joynes-Otto's  First  Book  in  French.     Boards 30  131 

Introductory  French  Lessons 100  131 

Introductory  French  Reader 80  131 

M^ras's  Syntaxe  Pratique  de  la  Langue  Fran9aise 100  132 

L^gendes   Fran^aises  :   No.  i.  Robert  le  Diable 20  132 

No.  2.  Le  Bon  Roi  Dagobert 20  132 

No.  3.  Merlin  TEnchanieur 30  132 

Moutonnier's  Les  Premiers  Pas  dans  T'fitude  du  Fran9ais 75  133 

Pour  Apprendre  £l  Parler  Fran9ais  75  133 

Otto's  French  Conversation-Grammar.     Half  roan.    (AVy,  60  cts.) 130  134 

Progressive  French  Reader •«... x  10  134 


Educational  Publications 


CATALOGUE 

PRICE  PAGB 

Parlez-vous  Fran9ais  ?     Boards $    40  134 

Pylodet's  Beginning  French.     Boards 45  13s 

Beginner's  French  Reader.    Boards 45  135 

Second  French  Reader 90  135 

Riodu's  Lucie 60  135 

Sadler's  Translating  English  into  French i  00  135 

Stern  and  Meras's  Etude  Progressive  de  la  Langue  Fran9aise i  20  136 

Whitney's  Practical  French  Grammar.     Half  roan.    (AVy,  80  cts.) 130  137 

Practical  French 90  138 

Brief  French  Grammar  65  239 

Introductory  French  Reader 70  140 

Witcomb  and  Bellenger's  Guide  to  French  Conversation 50  141 


FRENCH   LITERATURE. 

Achard's  Le  Clos  Pommier.    Paper 25  148 

The  same  with  De  Maistre's  Les  Prisonniers  du  Caucase 70  148 

^sop's  Fables  in  French 50  162 

Alliot's  Les  Auteurs  Contemporains 120  142 

Aubert's  Littdrature  Fran9aise i  00  142 

Balzac's  Eugenie  Grandet.     (Bergeron.) 80  231 

Bayard  et  Lemoine's  La  Niaise  de  Saint-Flour.    Playt    Paper 20  156 

B^doUiere's  Histoire  de  la  Mere  Michel.     Vocab 60  148 

The  same.    Paper 30  148 

Bishop's  Choy-Suzanne.     Boards 3°  232 

Carraud's  Les  Gouters  de  la  Grand'm^re.    Paper 20  162 

^\\.\s.^'^%\ix''^  Petites  Filles  Modeles 80  162 

Chateaubriand's  Les  Aventures  du  dernier  Abenc^rage.     With  extracts 
Uova.  Atala,  Voyage  en  Amerique^  t.ic.    (Sanderson.) 

Boards 35  233 

Choix  de  Contes  Contemporains.    (O'Connor.)     , 100  149 

The  same.    Paper  52  149 

Clairville's  Petites  Misbres  de  la  Vie  Humaine.    Play.    Paper 20  156 

Classic  French  Plays  : 

Vol.  L  Le  Cid,  Le  Misanthrope,  Athalie 100  145 

Vol.  IL  Cinna,  L'Avare,  Esther  100  145 

Vol.  IIL  Horace,  Bourgeois  Gentilhomme,  Les  Plaideurs i  00  145 

College  Series  of  French  Plays  : 

Vol.  L  Joie  fait  Peur,  Bataille  de  Dames,  Maison  de  Penarvan.  i  00  156 
Vol.  IL  Petits  Oiseaux,  Mile,  de  la  Seiglifere,  Roman  d'un  Jeune 

Homme  Pauvre,  Doigts  de  F^e 100  156 

Coraeille's  Cid.    (Joynes.)    Play.    Boards 20  145 

Cinna.    (Joynes.)    Play.     Boards 20  146 

Horace.    (Delbos.)    Play.    Boards 20  146 

euro's  La  Jeune  Savante,  with  Souvestre's  La  Loterie  de    Francfort. 

Plays.    Paper 20  160 

Daudet's  Contes.     Including  La  Belle  Nivernaise.     (Cameron.) 80  149 

La  Belle  Nivernaise.     (Cameron.)     Boards  25  149 

Drohojowska's  Demoiselle  de  Saint-Cyr.     With  Souvestre's  Testament 

de  Mme.  Patural.    Plays.    Boards 20  160 

De  Neuville's  Trois  Comedies  pour  Jeunts  Filles.     I.  Les  Cuisinieres. 

II.  Le  Petit  Tom.     III.  La  Malade  Imaginaire.     Paper..  35  162 

Erckmann-Chatrian's  Le  Conscrit  de  1813.    (Bocher.) 90  150 

The  same.     Boards 48  150 

Le  Blocus.     (Bocher.)  90  150 

The  same.     Paper 48  »So 

Madame  Th^rese.     (Bocher.) 90  150 

The  same.    Paper 48  150 

Pallet's  Les  Princes  de  1' Art i  00  150 

Thesame.    Paper 52  150 

Feuillet's  Le  Roman  d'un  Jeune  Homme  Pauvre.  The  Novel.  (Owen.)  90  151 

Thesame.     Paper •• 44  15* 

Le  Roman  d'un  Jeune  Homme  Pauvre.     The  Play.     Boards.  20  157 

L^  Village.     Play.    Paper  20  157 


viii  Complete  List  of  Henry  Holt  &  Co.'s 


CATALOGUE 

PRICE  PAGE 

F^val's  Chouans  et  Bleus.    (Sankey.) $    80  151 

The  same.     Paper 40  151 

Fleury's  L'Histoire  de  France  110  161 

Foa's  Le  Petit  Robinson  de  Paris.     Focai 70  151 

The  same.     Paper 36  151 

Contes  Biographiques.     Vocab 80  151 

The  same.     Paper 40  151 

Fortier's  Histoire  de  la  Litterature  Fran9aise  100  143 

Girardin's  La  Joie  fait  Peur.     Play.     Paper 20  157 

Halevy's  L'Abbe  Constantin.     Vocab.    (Super.)    Boards 40  233 

Hugo's  Selections.    (Warren.) 70  234 

Ruy  Bias.    Play.    (Michaels.)    Boards. 40  157 

Hernani.    Play.    (Harper.) 70  234 

Janon's  Recueil  de  Poesies 80  144 

Labiche  and  Delacour's  La  Cagnotte.     Play.    Paper 20  158 

Les  Petits  Oiseaux.     Play.     Paper 20  158 

Labiche  et  Martin's  La  Poudreaux  Yeux.     Play.     Paper 20  158 

Lacombe's  Petite  Histoire  du  Peuple  Franfais  60  161 

La  Fontaine's  Fables  Choisies.     (Delbos.)    Boards  40  146 

Leclerq's  Trois  Proverbes.     Plays.     Paper 20  158 

Mace's  Bouchee  de  Pain.     Vocab 100  152 

The  same.     Vocab.     Paper ,  52  152 

Madame  de  M.'s  La  Petite  Maman.  With  Mme.  de  Gaulle's  Le  Bracelet. 

Paper 20  162 

Mazeres' Le  Collier  de  Perles.     Play.     Paper 20  158 

de  Maistre's  Voyage  autour  de  ma  Chambre.     Paper 28  152 

Merimee's  Colomba.     (Cameron.) 60  237 

The  same.     Boards 36  237 

Moli^re's  L'Avare.     Play.    (Joynes.)    Boards 20  146 

Le  Bourgeois  Gentilhomme.    Play.    (Delbos.)     Paper 20  146 

Le  Misanthrope.     Play.    (Joynes.)    Boards 20  147 

Musiciens  C^l^bres  i  00  153 

The  same.    Paper 52  153 

Musset's  Un  Caprice.    Play.    Paper 20  158 

Porchat's  Trois  Mois  sous  la  Neige 70  153 

The  same.     Paper 32  153 

Pressense's  Rosa.     Vocab.    (Pylodet.) 100  154 

The  same.     Paper 52  154 

Pylodet's  Gouttes  de  Ros^e 50  144 

Le9ons  de  Litterature  Fran^atse  Classique 130  144 

Theatre  Fran9ais  Classique.     Paper 20  144 

La  Litterature  Fran9aise  Contemporaine i  10  144 

La  M^re  I'Oie.     Boards 40  163 

Racine's  Athalie.    Play.    (Joynes.)    Boards 20  147, 

Esther.    Play.    (Joynes.)     Boards 20  147 

Les  Plaideurs.    Play.    (Delbos.) 20  147 

Regent's  French  and  German  Poems.     Boards 20 

St.  Germain's  Pour  une  :fipingle.     Vocab 75  163 

The  same.     Paper 36  163 

Sand's  La  Petite  Fadette.     (BScher.) i  00  154 

The  same.     Boards 5a  154 

Marianne.    Paper  30  154 

Sandeau's  Mademoiselle  de  la  Seiglifere.     Play.     Boards  20  159 

La  Maison  de  Penarvan.     Play.     Boards 20  159 

Scribe  et  Legouve.     La  Bataille  de  Dames.     Play.     Boards 20  159 

Les  Doigts  de  Fee.    Play.    Boards ...   20  159 

Scribe  et  Melesville's  Valerie.    Play     Paper 20  159 

Segur's  Les  Petites  Filles  Modeles.     Paper 24  163 

Siraudin  et  Thiboust's  Les  Femmes  qui  Pleurent.    Play.     Paper 20  159 

Souvestre's  Un  Philosophe  sous  les  Toits  60  154 

The  same.    Paper 28  154 

La  Vieille  Cousine,  with  Les  Ricochets.     Plays.     Paper 20  160 

La  Loterie  de  Francfort,  with  Curo's  La  Jeune  Savante. 

Plays.     Boards 20  160 

Le  Testament  de  Mme.  Patural,  with  Drohojowska's  Demoi- 
selle de  Saint-Cyr.     Plays.     Boards 20  160 

Tvine's  Les  Origines  de  la  France  Coiiiemporaine.    (Edgren  )    Boards.  50  237 


Educational  Publications 


CATALOGUE 
PRICE    i'AGE 


Thiers'"  Expedition  de  Bonaparte  en  figypte.     (Rdgren.)     Boards $  35  238 

ToepfiFer's  Bibliothfeque  de  mon  Oncie.     (Marcou.)   238 

Vacquerie's  Jean  Baud ry,     Piay.     Paper 20  160 

Verconsin's  C'fiiait  Gertrude.   En  Wagon.    (.Together.)   Plays.    Boards.  30  23S 

Verne's  Michel  Strogoff.     (.Lewis.) 70  155 

Walter's  Classic  French  Letters 75  239 


GREEK  AND   LATIN. 

Brooks's  Introduction  to  Attic  Greek i  10  164 

Goodell's  The  Greek  in  English  60  165 

Greek  Lessons.     Part  I.  The  Greek  in  English.     Part  IL  The 

Greek  of  Xenophon   125  166 

Judson's  The  Latin  in  English     243 

Peck's  Gai  Suetoni  Tranquilli  De  Vila  Caesarum  Libri  Duo i  20  167 

Lati  n  Pronunciation ...  40  167 

Preparatory  Latin  and  Greek  Texts  120  168 

Latin  part  separate 80  168 

Greek  part  separate 60  168 

Richardson's  Six  Months'  Preparation  for  Caesar 90  245 

Scrivener's  Greek  Testament  200  168 

Williams's  Extracts  from  Various  Greek   Authors i  cxj  169 

ITALIAN    AND   SPANISH. 

ITALIAN 

Montague's  Manual  of  Italian  Grammar.     Half  roan 100  171 

Nota's  La  Fiera.     Paper 60  173 

Ongaro's  Rosa  deir  Alpi.    Paper.   60  173 

Parlate  Italiano  ?     Boards   40  173 

Pellico's  Francesca  da  Rimini.     Paper 60  173 

SPANISH. 

Caballero''s  La  Familia  de  Alvareda.     Paper -5  173 

i  Habla  vd.  Espanol  ?     Boards ^^ 40  172 

£  Habla  V.  Ingles  ?     Boards .    .. 40  172 

Lope  de  Vega's  Obras  Maestras.     Burnished  buckram 100  173 

Manning's  Practical  Spanish  Grammar.     (Revised  Ed.) 100  170 

Ramsey's  Text-book  of  Modern  Spanish 180  172 

Saies's  Spanish  Hive , , i  c»  172 


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